| L(s) = 1 | + 3-s + 3·5-s − 2·9-s − 3·11-s + 2·13-s + 3·15-s + 3·17-s − 19-s + 3·23-s + 4·25-s − 5·27-s − 6·29-s − 7·31-s − 3·33-s − 37-s + 2·39-s + 6·41-s − 4·43-s − 6·45-s − 9·47-s + 3·51-s + 3·53-s − 9·55-s − 57-s + 9·59-s − 61-s + 6·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s − 2/3·9-s − 0.904·11-s + 0.554·13-s + 0.774·15-s + 0.727·17-s − 0.229·19-s + 0.625·23-s + 4/5·25-s − 0.962·27-s − 1.11·29-s − 1.25·31-s − 0.522·33-s − 0.164·37-s + 0.320·39-s + 0.937·41-s − 0.609·43-s − 0.894·45-s − 1.31·47-s + 0.420·51-s + 0.412·53-s − 1.21·55-s − 0.132·57-s + 1.17·59-s − 0.128·61-s + 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.567274972\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.567274972\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 + 7 T + p T^{2} \) | 1.67.h |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + T + p T^{2} \) | 1.73.b |
| 79 | \( 1 + 13 T + p T^{2} \) | 1.79.n |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 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
−12.92295671713312891956034975071, −11.38571601734354388504430939305, −10.41653304867165853405717544740, −9.465516888415029842549770658404, −8.646784595072977282306137209646, −7.51155146227211557820915407148, −6.02455306491682974275355684196, −5.28434328079604079625535316421, −3.31407617989174490462667281240, −2.04448974040096779312937543400,
2.04448974040096779312937543400, 3.31407617989174490462667281240, 5.28434328079604079625535316421, 6.02455306491682974275355684196, 7.51155146227211557820915407148, 8.646784595072977282306137209646, 9.465516888415029842549770658404, 10.41653304867165853405717544740, 11.38571601734354388504430939305, 12.92295671713312891956034975071
