| L(s) = 1 | + 6·2-s + 4·3-s + 9·4-s − 25·5-s + 24·6-s − 3·7-s + 12·8-s − 21·9-s − 150·10-s + 23·11-s + 36·12-s + 132·13-s − 18·14-s − 100·15-s + 139·16-s + 23·17-s − 126·18-s − 161·19-s − 225·20-s − 12·21-s + 138·22-s − 115·23-s + 48·24-s + 375·25-s + 792·26-s + 51·27-s − 27·28-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 0.769·3-s + 9/8·4-s − 2.23·5-s + 1.63·6-s − 0.161·7-s + 0.530·8-s − 7/9·9-s − 4.74·10-s + 0.630·11-s + 0.866·12-s + 2.81·13-s − 0.343·14-s − 1.72·15-s + 2.17·16-s + 0.328·17-s − 1.64·18-s − 1.94·19-s − 2.51·20-s − 0.124·21-s + 1.33·22-s − 1.04·23-s + 0.408·24-s + 3·25-s + 5.97·26-s + 0.363·27-s − 0.182·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{5} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{5} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(10.67595434\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.67595434\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( ( 1 + p T )^{5} \) |
| 23 | $C_1$ | \( ( 1 + p T )^{5} \) |
| good | 2 | $C_2 \wr S_5$ | \( 1 - 3 p T + 27 T^{2} - 15 p^{3} T^{3} + 205 p T^{4} - 299 p^{2} T^{5} + 205 p^{4} T^{6} - 15 p^{9} T^{7} + 27 p^{9} T^{8} - 3 p^{13} T^{9} + p^{15} T^{10} \) |
| 3 | $C_2 \wr S_5$ | \( 1 - 4 T + 37 T^{2} - 283 T^{3} + 1888 T^{4} - 5770 T^{5} + 1888 p^{3} T^{6} - 283 p^{6} T^{7} + 37 p^{9} T^{8} - 4 p^{12} T^{9} + p^{15} T^{10} \) |
| 7 | $C_2 \wr S_5$ | \( 1 + 3 T + 1231 T^{2} + 839 p T^{3} + 712735 T^{4} + 454030 p T^{5} + 712735 p^{3} T^{6} + 839 p^{7} T^{7} + 1231 p^{9} T^{8} + 3 p^{12} T^{9} + p^{15} T^{10} \) |
| 11 | $C_2 \wr S_5$ | \( 1 - 23 T + 240 p T^{2} - 12088 T^{3} + 4408887 T^{4} - 24823298 T^{5} + 4408887 p^{3} T^{6} - 12088 p^{6} T^{7} + 240 p^{10} T^{8} - 23 p^{12} T^{9} + p^{15} T^{10} \) |
| 13 | $C_2 \wr S_5$ | \( 1 - 132 T + 15771 T^{2} - 1183283 T^{3} + 79334508 T^{4} - 3923517810 T^{5} + 79334508 p^{3} T^{6} - 1183283 p^{6} T^{7} + 15771 p^{9} T^{8} - 132 p^{12} T^{9} + p^{15} T^{10} \) |
| 17 | $C_2 \wr S_5$ | \( 1 - 23 T + 9933 T^{2} - 78943 T^{3} + 66831939 T^{4} - 704411084 T^{5} + 66831939 p^{3} T^{6} - 78943 p^{6} T^{7} + 9933 p^{9} T^{8} - 23 p^{12} T^{9} + p^{15} T^{10} \) |
| 19 | $C_2 \wr S_5$ | \( 1 + 161 T + 38294 T^{2} + 4388196 T^{3} + 552284485 T^{4} + 45047697766 T^{5} + 552284485 p^{3} T^{6} + 4388196 p^{6} T^{7} + 38294 p^{9} T^{8} + 161 p^{12} T^{9} + p^{15} T^{10} \) |
| 29 | $C_2 \wr S_5$ | \( 1 - 401 T + 142982 T^{2} - 31664487 T^{3} + 6793391111 T^{4} - 1073631941944 T^{5} + 6793391111 p^{3} T^{6} - 31664487 p^{6} T^{7} + 142982 p^{9} T^{8} - 401 p^{12} T^{9} + p^{15} T^{10} \) |
| 31 | $C_2 \wr S_5$ | \( 1 - 32 T + 4166 p T^{2} - 2351709 T^{3} + 7088287729 T^{4} - 83757362501 T^{5} + 7088287729 p^{3} T^{6} - 2351709 p^{6} T^{7} + 4166 p^{10} T^{8} - 32 p^{12} T^{9} + p^{15} T^{10} \) |
| 37 | $C_2 \wr S_5$ | \( 1 + 38 T + 49418 T^{2} - 13674884 T^{3} + 5128352681 T^{4} + 10357772588 T^{5} + 5128352681 p^{3} T^{6} - 13674884 p^{6} T^{7} + 49418 p^{9} T^{8} + 38 p^{12} T^{9} + p^{15} T^{10} \) |
| 41 | $C_2 \wr S_5$ | \( 1 + 12 T + 164750 T^{2} - 2108855 T^{3} + 13077705771 T^{4} - 290575261939 T^{5} + 13077705771 p^{3} T^{6} - 2108855 p^{6} T^{7} + 164750 p^{9} T^{8} + 12 p^{12} T^{9} + p^{15} T^{10} \) |
| 43 | $C_2 \wr S_5$ | \( 1 + 566 T + 382455 T^{2} + 141471144 T^{3} + 1359815254 p T^{4} + 15844894402212 T^{5} + 1359815254 p^{4} T^{6} + 141471144 p^{6} T^{7} + 382455 p^{9} T^{8} + 566 p^{12} T^{9} + p^{15} T^{10} \) |
| 47 | $C_2 \wr S_5$ | \( 1 - 919 T + 755197 T^{2} - 395537760 T^{3} + 3829809196 p T^{4} - 62201848223442 T^{5} + 3829809196 p^{4} T^{6} - 395537760 p^{6} T^{7} + 755197 p^{9} T^{8} - 919 p^{12} T^{9} + p^{15} T^{10} \) |
| 53 | $C_2 \wr S_5$ | \( 1 - 1156 T + 952322 T^{2} - 515125430 T^{3} + 246241392685 T^{4} - 96416322127068 T^{5} + 246241392685 p^{3} T^{6} - 515125430 p^{6} T^{7} + 952322 p^{9} T^{8} - 1156 p^{12} T^{9} + p^{15} T^{10} \) |
| 59 | $C_2 \wr S_5$ | \( 1 - 1324 T + 846904 T^{2} - 218423774 T^{3} - 30078279353 T^{4} + 46254390447908 T^{5} - 30078279353 p^{3} T^{6} - 218423774 p^{6} T^{7} + 846904 p^{9} T^{8} - 1324 p^{12} T^{9} + p^{15} T^{10} \) |
| 61 | $C_2 \wr S_5$ | \( 1 + 1673 T + 1673514 T^{2} + 1161920040 T^{3} + 650056213753 T^{4} + 320984537769678 T^{5} + 650056213753 p^{3} T^{6} + 1161920040 p^{6} T^{7} + 1673514 p^{9} T^{8} + 1673 p^{12} T^{9} + p^{15} T^{10} \) |
| 67 | $C_2 \wr S_5$ | \( 1 - 558 T + 1591744 T^{2} - 674651764 T^{3} + 983770331347 T^{4} - 304863378049148 T^{5} + 983770331347 p^{3} T^{6} - 674651764 p^{6} T^{7} + 1591744 p^{9} T^{8} - 558 p^{12} T^{9} + p^{15} T^{10} \) |
| 71 | $C_2 \wr S_5$ | \( 1 + 108 T + 718176 T^{2} - 229462863 T^{3} + 199089282365 T^{4} - 176285333941687 T^{5} + 199089282365 p^{3} T^{6} - 229462863 p^{6} T^{7} + 718176 p^{9} T^{8} + 108 p^{12} T^{9} + p^{15} T^{10} \) |
| 73 | $C_2 \wr S_5$ | \( 1 - 1173 T + 1191083 T^{2} - 1026657468 T^{3} + 769848826316 T^{4} - 544640203216094 T^{5} + 769848826316 p^{3} T^{6} - 1026657468 p^{6} T^{7} + 1191083 p^{9} T^{8} - 1173 p^{12} T^{9} + p^{15} T^{10} \) |
| 79 | $C_2 \wr S_5$ | \( 1 - 656 T + 1078459 T^{2} - 469459264 T^{3} + 567047460634 T^{4} - 234474312862816 T^{5} + 567047460634 p^{3} T^{6} - 469459264 p^{6} T^{7} + 1078459 p^{9} T^{8} - 656 p^{12} T^{9} + p^{15} T^{10} \) |
| 83 | $C_2 \wr S_5$ | \( 1 + 82 T + 1583116 T^{2} - 445539716 T^{3} + 1062198058723 T^{4} - 544818537547324 T^{5} + 1062198058723 p^{3} T^{6} - 445539716 p^{6} T^{7} + 1583116 p^{9} T^{8} + 82 p^{12} T^{9} + p^{15} T^{10} \) |
| 89 | $C_2 \wr S_5$ | \( 1 - 570 T + 1395669 T^{2} - 16135144 T^{3} + 622215877666 T^{4} + 428704319615588 T^{5} + 622215877666 p^{3} T^{6} - 16135144 p^{6} T^{7} + 1395669 p^{9} T^{8} - 570 p^{12} T^{9} + p^{15} T^{10} \) |
| 97 | $C_2 \wr S_5$ | \( 1 - 633 T + 2758024 T^{2} - 1209150498 T^{3} + 4197884554679 T^{4} - 1633564634251466 T^{5} + 4197884554679 p^{3} T^{6} - 1209150498 p^{6} T^{7} + 2758024 p^{9} T^{8} - 633 p^{12} T^{9} + p^{15} T^{10} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.174882123397384072111575886134, −7.890627998094180743156842988206, −7.55812410833966732413803671307, −7.54906478306587026346711779162, −6.94472487130905508025393066757, −6.85316180382623169004549196540, −6.45859579729192296268086441367, −6.16502609749531657616849402475, −6.08800160953761094023893853908, −5.99402405661476908645184839817, −5.22482506741171296884994348225, −5.04726139812454266012391483942, −4.80359286017242429493817590716, −4.55900292119228544154524619392, −4.40349074846768530393434012649, −3.95782787608750176330015477855, −3.88885640964164232945636930315, −3.50061766623195554083395588205, −3.44421399713674782559826242602, −3.29939336007389935685831012019, −2.55568974584814226338514522095, −2.19895551819804108901185471354, −1.48234843500660029761789802229, −0.843850271307433494817457492963, −0.59393817036756763483328392026,
0.59393817036756763483328392026, 0.843850271307433494817457492963, 1.48234843500660029761789802229, 2.19895551819804108901185471354, 2.55568974584814226338514522095, 3.29939336007389935685831012019, 3.44421399713674782559826242602, 3.50061766623195554083395588205, 3.88885640964164232945636930315, 3.95782787608750176330015477855, 4.40349074846768530393434012649, 4.55900292119228544154524619392, 4.80359286017242429493817590716, 5.04726139812454266012391483942, 5.22482506741171296884994348225, 5.99402405661476908645184839817, 6.08800160953761094023893853908, 6.16502609749531657616849402475, 6.45859579729192296268086441367, 6.85316180382623169004549196540, 6.94472487130905508025393066757, 7.54906478306587026346711779162, 7.55812410833966732413803671307, 7.890627998094180743156842988206, 8.174882123397384072111575886134
