| L(s) = 1 | − 6·2-s + 10·3-s + 4·4-s − 72·5-s − 60·6-s − 196·7-s + 168·8-s − 143·9-s + 432·10-s + 450·11-s + 40·12-s − 142·13-s + 1.17e3·14-s − 720·15-s − 1.13e3·16-s − 289·17-s + 858·18-s − 244·19-s − 288·20-s − 1.96e3·21-s − 2.70e3·22-s + 2.90e3·23-s + 1.68e3·24-s + 2.05e3·25-s + 852·26-s − 3.86e3·27-s − 784·28-s + ⋯ |
| L(s) = 1 | − 1.06·2-s + 0.641·3-s + 1/8·4-s − 1.28·5-s − 0.680·6-s − 1.51·7-s + 0.928·8-s − 0.588·9-s + 1.36·10-s + 1.12·11-s + 0.0801·12-s − 0.233·13-s + 1.60·14-s − 0.826·15-s − 1.10·16-s − 0.242·17-s + 0.624·18-s − 0.155·19-s − 0.160·20-s − 0.969·21-s − 1.18·22-s + 1.14·23-s + 0.595·24-s + 0.658·25-s + 0.247·26-s − 1.01·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + p^{2} T \) |
| good | 2 | \( 1 + 3 p T + p^{5} T^{2} \) |
| 3 | \( 1 - 10 T + p^{5} T^{2} \) |
| 5 | \( 1 + 72 T + p^{5} T^{2} \) |
| 7 | \( 1 + 4 p^{2} T + p^{5} T^{2} \) |
| 11 | \( 1 - 450 T + p^{5} T^{2} \) |
| 13 | \( 1 + 142 T + p^{5} T^{2} \) |
| 19 | \( 1 + 244 T + p^{5} T^{2} \) |
| 23 | \( 1 - 2904 T + p^{5} T^{2} \) |
| 29 | \( 1 + 6984 T + p^{5} T^{2} \) |
| 31 | \( 1 + 436 T + p^{5} T^{2} \) |
| 37 | \( 1 + 8572 T + p^{5} T^{2} \) |
| 41 | \( 1 - 16374 T + p^{5} T^{2} \) |
| 43 | \( 1 + 19216 T + p^{5} T^{2} \) |
| 47 | \( 1 + 19920 T + p^{5} T^{2} \) |
| 53 | \( 1 - 1146 T + p^{5} T^{2} \) |
| 59 | \( 1 - 22008 T + p^{5} T^{2} \) |
| 61 | \( 1 - 35780 T + p^{5} T^{2} \) |
| 67 | \( 1 - 23264 T + p^{5} T^{2} \) |
| 71 | \( 1 + 31704 T + p^{5} T^{2} \) |
| 73 | \( 1 + 13966 T + p^{5} T^{2} \) |
| 79 | \( 1 + 51088 T + p^{5} T^{2} \) |
| 83 | \( 1 + 64344 T + p^{5} T^{2} \) |
| 89 | \( 1 - 70650 T + p^{5} T^{2} \) |
| 97 | \( 1 - 62702 T + p^{5} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.13135021049851787741106515416, −16.16326424319327714410639986222, −14.76126956354781260992680504910, −13.08654043182536750617591410850, −11.39517673836485203095439653398, −9.552622313961293803852960332341, −8.603629201181327167674987462957, −7.09924914425855796890457904527, −3.63189009343879270910940266258, 0,
3.63189009343879270910940266258, 7.09924914425855796890457904527, 8.603629201181327167674987462957, 9.552622313961293803852960332341, 11.39517673836485203095439653398, 13.08654043182536750617591410850, 14.76126956354781260992680504910, 16.16326424319327714410639986222, 17.13135021049851787741106515416
