| L(s) = 1 | + 3·5-s − 7-s − 3·11-s − 13-s − 17-s − 3·19-s + 5·23-s + 4·25-s + 3·29-s + 8·31-s − 3·35-s − 5·37-s − 12·41-s + 7·43-s + 8·47-s + 49-s + 2·53-s − 9·55-s + 61-s − 3·65-s − 14·67-s − 4·71-s − 11·73-s + 3·77-s − 6·79-s + 6·83-s − 3·85-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 0.377·7-s − 0.904·11-s − 0.277·13-s − 0.242·17-s − 0.688·19-s + 1.04·23-s + 4/5·25-s + 0.557·29-s + 1.43·31-s − 0.507·35-s − 0.821·37-s − 1.87·41-s + 1.06·43-s + 1.16·47-s + 1/7·49-s + 0.274·53-s − 1.21·55-s + 0.128·61-s − 0.372·65-s − 1.71·67-s − 0.474·71-s − 1.28·73-s + 0.341·77-s − 0.675·79-s + 0.658·83-s − 0.325·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 \) | |
| 3 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−15.60171059848442, −15.03561777066498, −14.50660233662055, −13.69302756789993, −13.52315872969436, −13.10736131495274, −12.35423169122587, −12.01687556586245, −11.09786074462593, −10.41757319618128, −10.28792642253576, −9.674770722012977, −8.960329761047133, −8.614144220124274, −7.869257455941522, −6.994373835704634, −6.714426009721568, −5.883444502656811, −5.538629159987257, −4.803030800641744, −4.254313512418769, −3.071174602324820, −2.702982556889466, −1.988257740670940, −1.149287217943878, 0,
1.149287217943878, 1.988257740670940, 2.702982556889466, 3.071174602324820, 4.254313512418769, 4.803030800641744, 5.538629159987257, 5.883444502656811, 6.714426009721568, 6.994373835704634, 7.869257455941522, 8.614144220124274, 8.960329761047133, 9.674770722012977, 10.28792642253576, 10.41757319618128, 11.09786074462593, 12.01687556586245, 12.35423169122587, 13.10736131495274, 13.52315872969436, 13.69302756789993, 14.50660233662055, 15.03561777066498, 15.60171059848442
