| L(s) = 1 | + 4·2-s + 5·3-s + 12·4-s + 20·6-s + 32·8-s + 9·9-s − 5·11-s + 60·12-s + 95·13-s + 80·16-s − 25·17-s + 36·18-s + 120·19-s − 20·22-s − 124·23-s + 160·24-s + 380·26-s + 100·27-s + 213·29-s − 70·31-s + 192·32-s − 25·33-s − 100·34-s + 108·36-s + 134·37-s + 480·38-s + 475·39-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.962·3-s + 3/2·4-s + 1.36·6-s + 1.41·8-s + 1/3·9-s − 0.137·11-s + 1.44·12-s + 2.02·13-s + 5/4·16-s − 0.356·17-s + 0.471·18-s + 1.44·19-s − 0.193·22-s − 1.12·23-s + 1.36·24-s + 2.86·26-s + 0.712·27-s + 1.36·29-s − 0.405·31-s + 1.06·32-s − 0.131·33-s − 0.504·34-s + 1/2·36-s + 0.595·37-s + 2.04·38-s + 1.95·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(21.97091094\) |
| \(L(\frac12)\) |
\(\approx\) |
\(21.97091094\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - 5 T + 16 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 5 T + 142 p T^{2} + 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 95 T + 6252 T^{2} - 95 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 25 T + 9938 T^{2} + 25 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 120 T + 16610 T^{2} - 120 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 124 T + 10478 T^{2} + 124 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 213 T + 32464 T^{2} - 213 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 70 T + 60630 T^{2} + 70 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 134 T + 65970 T^{2} - 134 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 370 T + 108170 T^{2} + 370 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 82 T + 156270 T^{2} - 82 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 115 T - 93886 T^{2} - 115 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 1024 T + 542198 T^{2} - 1024 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 760 T + 497810 T^{2} + 760 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 790 T + 417234 T^{2} - 790 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 850 T + 777726 T^{2} + 850 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 372 T + 307918 T^{2} - 372 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 190 T + 289866 T^{2} - 190 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 425 T + 897378 T^{2} - 425 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2110 T + 2255006 T^{2} - 2110 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 560 T + 355538 T^{2} + 560 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 675 T + 1619546 T^{2} - 675 p^{3} T^{3} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.580661842740645646442048548231, −8.516518034139553370134228477022, −7.87819101150490422710895809210, −7.67969185244640026696599676237, −7.27134274793594105545128925930, −6.63913253625871484261090525462, −6.30532889994753007849620403296, −6.26359188162475253079523053182, −5.39856575053626602355381398256, −5.39128544902821324798474102290, −4.80506885578912858051965636848, −4.23619184706375248501710678050, −3.77791843315546404796021204364, −3.71446746885510555592058397356, −3.03497159101190926884926513658, −2.83755031320259046990914659230, −2.21201117569832499318439089985, −1.74216875926491771284938521404, −1.10004779568060768161693051758, −0.69408018532467909109180801839,
0.69408018532467909109180801839, 1.10004779568060768161693051758, 1.74216875926491771284938521404, 2.21201117569832499318439089985, 2.83755031320259046990914659230, 3.03497159101190926884926513658, 3.71446746885510555592058397356, 3.77791843315546404796021204364, 4.23619184706375248501710678050, 4.80506885578912858051965636848, 5.39128544902821324798474102290, 5.39856575053626602355381398256, 6.26359188162475253079523053182, 6.30532889994753007849620403296, 6.63913253625871484261090525462, 7.27134274793594105545128925930, 7.67969185244640026696599676237, 7.87819101150490422710895809210, 8.516518034139553370134228477022, 8.580661842740645646442048548231
