| L(s) = 1 | + 4·2-s + 3-s + 10·4-s + 3·5-s + 4·6-s − 7-s + 20·8-s − 3·9-s + 12·10-s + 10·12-s − 9·13-s − 4·14-s + 3·15-s + 35·16-s − 2·17-s − 12·18-s − 4·19-s + 30·20-s − 21-s − 2·23-s + 20·24-s + 2·25-s − 36·26-s − 5·27-s − 10·28-s + 5·29-s + 12·30-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 0.577·3-s + 5·4-s + 1.34·5-s + 1.63·6-s − 0.377·7-s + 7.07·8-s − 9-s + 3.79·10-s + 2.88·12-s − 2.49·13-s − 1.06·14-s + 0.774·15-s + 35/4·16-s − 0.485·17-s − 2.82·18-s − 0.917·19-s + 6.70·20-s − 0.218·21-s − 0.417·23-s + 4.08·24-s + 2/5·25-s − 7.06·26-s − 0.962·27-s − 1.88·28-s + 0.928·29-s + 2.19·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 11^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 11^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(59.44815932\) |
| \(L(\frac12)\) |
\(\approx\) |
\(59.44815932\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 11 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 3 | $C_2 \wr S_4$ | \( 1 - T + 4 T^{2} - 2 T^{3} + 5 T^{4} - 2 p T^{5} + 4 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 - 3 T + 7 T^{2} - 9 T^{3} + 16 T^{4} - 9 p T^{5} + 7 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 + T + 18 T^{2} + 2 T^{3} + 3 p^{2} T^{4} + 2 p T^{5} + 18 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 9 T + 66 T^{2} + 336 T^{3} + 105 p T^{4} + 336 p T^{5} + 66 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 2 T + 49 T^{2} + 64 T^{3} + 1116 T^{4} + 64 p T^{5} + 49 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 2 T + 77 T^{2} + 118 T^{3} + 2488 T^{4} + 118 p T^{5} + 77 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 5 T + 30 T^{2} - 216 T^{3} + 2081 T^{4} - 216 p T^{5} + 30 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 27 T + 373 T^{2} - 3367 T^{3} + 21848 T^{4} - 3367 p T^{5} + 373 p^{2} T^{6} - 27 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 6 T + 80 T^{2} - 442 T^{3} + 3054 T^{4} - 442 p T^{5} + 80 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 5 T + 91 T^{2} + 463 T^{3} + 4896 T^{4} + 463 p T^{5} + 91 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - T + 107 T^{2} + 179 T^{3} + 5112 T^{4} + 179 p T^{5} + 107 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 22 T + 304 T^{2} - 2942 T^{3} + 22750 T^{4} - 2942 p T^{5} + 304 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 105 T^{2} + 564 T^{3} + 4728 T^{4} + 564 p T^{5} + 105 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 157 T^{2} + 72 T^{3} + 12652 T^{4} + 72 p T^{5} + 157 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 6 T + 216 T^{2} - 18 p T^{3} + 18942 T^{4} - 18 p^{2} T^{5} + 216 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 17 T + 326 T^{2} - 3450 T^{3} + 34683 T^{4} - 3450 p T^{5} + 326 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 19 T + 387 T^{2} + 4167 T^{3} + 44900 T^{4} + 4167 p T^{5} + 387 p^{2} T^{6} + 19 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 20 T + 413 T^{2} - 4592 T^{3} + 49752 T^{4} - 4592 p T^{5} + 413 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 12 T + 208 T^{2} - 1324 T^{3} + 17918 T^{4} - 1324 p T^{5} + 208 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 23 T + 375 T^{2} + 4591 T^{3} + 44424 T^{4} + 4591 p T^{5} + 375 p^{2} T^{6} + 23 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 16 T + 288 T^{2} - 2816 T^{3} + 34686 T^{4} - 2816 p T^{5} + 288 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 8 T + 172 T^{2} - 1720 T^{3} + 25174 T^{4} - 1720 p T^{5} + 172 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.91763715514310266135002897183, −5.50649730685322706967605920207, −5.50585801869072695852639885008, −5.24611987409778721049055125504, −5.04503375909564258993087904838, −4.74705619360292064878224705032, −4.66291474708359697863926335738, −4.62892574496606646168285254696, −4.48612253371877496917815690897, −4.13604266717085595318404063775, −3.92506277376722853497997564478, −3.67062036141611276731358031056, −3.51452780774816815635101506995, −3.05876019671653571166033312763, −2.98702723975832688492985039688, −2.91163334163347272465203448242, −2.40829878596657824325837787865, −2.39449014334551195004604322270, −2.35287060438431455140103728894, −2.29284464026367898102231178250, −1.95217865451676844045599765259, −1.38658662756059912951491571656, −1.23373707400815539812556394615, −0.62266507453428662005902884984, −0.50957847393700826900436567101,
0.50957847393700826900436567101, 0.62266507453428662005902884984, 1.23373707400815539812556394615, 1.38658662756059912951491571656, 1.95217865451676844045599765259, 2.29284464026367898102231178250, 2.35287060438431455140103728894, 2.39449014334551195004604322270, 2.40829878596657824325837787865, 2.91163334163347272465203448242, 2.98702723975832688492985039688, 3.05876019671653571166033312763, 3.51452780774816815635101506995, 3.67062036141611276731358031056, 3.92506277376722853497997564478, 4.13604266717085595318404063775, 4.48612253371877496917815690897, 4.62892574496606646168285254696, 4.66291474708359697863926335738, 4.74705619360292064878224705032, 5.04503375909564258993087904838, 5.24611987409778721049055125504, 5.50585801869072695852639885008, 5.50649730685322706967605920207, 5.91763715514310266135002897183
