| L(s) = 1 | − 4·2-s + 12·4-s + 12·5-s − 32·8-s − 48·10-s − 4·11-s − 48·13-s + 80·16-s + 132·17-s − 120·19-s + 144·20-s + 16·22-s + 76·23-s − 140·25-s + 192·26-s + 112·29-s − 432·31-s − 192·32-s − 528·34-s − 280·37-s + 480·38-s − 384·40-s + 36·41-s − 128·43-s − 48·44-s − 304·46-s − 264·47-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 1.07·5-s − 1.41·8-s − 1.51·10-s − 0.109·11-s − 1.02·13-s + 5/4·16-s + 1.88·17-s − 1.44·19-s + 1.60·20-s + 0.155·22-s + 0.689·23-s − 1.11·25-s + 1.44·26-s + 0.717·29-s − 2.50·31-s − 1.06·32-s − 2.66·34-s − 1.24·37-s + 2.04·38-s − 1.51·40-s + 0.137·41-s − 0.453·43-s − 0.164·44-s − 0.974·46-s − 0.819·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 - 12 T + 284 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 2594 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 48 T + 4520 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 132 T + 820 p T^{2} - 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 120 T + 13086 T^{2} + 120 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 76 T + 4970 T^{2} - 76 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 112 T + 6914 T^{2} - 112 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 432 T + 100406 T^{2} + 432 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 280 T + 56106 T^{2} + 280 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 36 T + 105908 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 128 T - 2778 T^{2} + 128 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 264 T - 3418 T^{2} + 264 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 268 T + 241982 T^{2} + 268 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 336 T + 261374 T^{2} - 336 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 504 T + 268248 T^{2} - 504 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 384 T + 596918 T^{2} + 384 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 396 T + 521098 T^{2} - 396 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 312 T + 94320 T^{2} - 312 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 848 T + 985854 T^{2} + 848 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 648 T + 1235750 T^{2} + 648 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 612 T + 1442324 T^{2} + 612 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2184 T + 2982432 T^{2} - 2184 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.599565989586647906717744167688, −9.276863102531705314321616082738, −8.671245539802345646235984955939, −8.451148238818894511191537357510, −7.74620278928510678559336029065, −7.66363586644873349491106758853, −6.88483501170140171596040817226, −6.87040927879460826219472653087, −5.92826087059954899252565624188, −5.91617367603843739799934863662, −5.09612410161413083304264282038, −5.05225765678717653038698737627, −3.69210546625416640933482738414, −3.68288457734591161389460733912, −2.50146362503905051724246523165, −2.46067918292872498821389127206, −1.47561082680756432879192476242, −1.43187017613334314839745591501, 0, 0,
1.43187017613334314839745591501, 1.47561082680756432879192476242, 2.46067918292872498821389127206, 2.50146362503905051724246523165, 3.68288457734591161389460733912, 3.69210546625416640933482738414, 5.05225765678717653038698737627, 5.09612410161413083304264282038, 5.91617367603843739799934863662, 5.92826087059954899252565624188, 6.87040927879460826219472653087, 6.88483501170140171596040817226, 7.66363586644873349491106758853, 7.74620278928510678559336029065, 8.451148238818894511191537357510, 8.671245539802345646235984955939, 9.276863102531705314321616082738, 9.599565989586647906717744167688
