| L(s) = 1 | + 3-s + 5-s + 5·7-s − 2·9-s − 5·11-s + 15-s − 17-s − 3·19-s + 5·21-s − 3·23-s + 25-s − 5·27-s − 29-s − 5·33-s + 5·35-s − 7·37-s + 5·41-s − 5·43-s − 2·45-s + 12·47-s + 18·49-s − 51-s + 2·53-s − 5·55-s − 3·57-s − 11·59-s − 13·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.88·7-s − 2/3·9-s − 1.50·11-s + 0.258·15-s − 0.242·17-s − 0.688·19-s + 1.09·21-s − 0.625·23-s + 1/5·25-s − 0.962·27-s − 0.185·29-s − 0.870·33-s + 0.845·35-s − 1.15·37-s + 0.780·41-s − 0.762·43-s − 0.298·45-s + 1.75·47-s + 18/7·49-s − 0.140·51-s + 0.274·53-s − 0.674·55-s − 0.397·57-s − 1.43·59-s − 1.66·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 7 | \( 1 - 5 T + p T^{2} \) | 1.7.af |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 11 T + p T^{2} \) | 1.59.l |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 - 3 T + p T^{2} \) | 1.67.ad |
| 71 | \( 1 - 13 T + p T^{2} \) | 1.71.an |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 7 T + p T^{2} \) | 1.89.h |
| 97 | \( 1 + 11 T + p T^{2} \) | 1.97.l |
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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
−16.70536780467493, −15.60849890095711, −15.30740596613991, −14.81302743928754, −14.10135036420487, −13.83033208330887, −13.39131574934631, −12.42366673760657, −12.06912929022746, −11.11085832932141, −10.83935634901124, −10.39481861060789, −9.449461151548710, −8.799788513883745, −8.316558294647184, −7.839757428514685, −7.422195246526202, −6.345909810698557, −5.542785689861175, −5.178372220270951, −4.495015474378252, −3.669334658949759, −2.527590323685143, −2.280708558558880, −1.419065693240272, 0,
1.419065693240272, 2.280708558558880, 2.527590323685143, 3.669334658949759, 4.495015474378252, 5.178372220270951, 5.542785689861175, 6.345909810698557, 7.422195246526202, 7.839757428514685, 8.316558294647184, 8.799788513883745, 9.449461151548710, 10.39481861060789, 10.83935634901124, 11.11085832932141, 12.06912929022746, 12.42366673760657, 13.39131574934631, 13.83033208330887, 14.10135036420487, 14.81302743928754, 15.30740596613991, 15.60849890095711, 16.70536780467493
