| L(s) = 1 | − 2-s + 4-s + 2·5-s + 4·7-s − 8-s − 2·10-s + 4·11-s + 2·13-s − 4·14-s + 16-s − 2·17-s − 4·19-s + 2·20-s − 4·22-s − 25-s − 2·26-s + 4·28-s + 6·29-s − 8·31-s − 32-s + 2·34-s + 8·35-s − 2·37-s + 4·38-s − 2·40-s − 41-s + 4·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s + 1.51·7-s − 0.353·8-s − 0.632·10-s + 1.20·11-s + 0.554·13-s − 1.06·14-s + 1/4·16-s − 0.485·17-s − 0.917·19-s + 0.447·20-s − 0.852·22-s − 1/5·25-s − 0.392·26-s + 0.755·28-s + 1.11·29-s − 1.43·31-s − 0.176·32-s + 0.342·34-s + 1.35·35-s − 0.328·37-s + 0.648·38-s − 0.316·40-s − 0.156·41-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.597776839\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.597776839\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 41 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−10.39359408099131626317482013831, −9.383700722197999637154978992730, −8.713426150404355454628361302705, −8.042238744681496704727000706394, −6.86414604779547030910547339232, −6.12115484502975746031593422200, −5.04471008961793371383618088791, −3.91210488051277779320076234638, −2.14898913496092963575229255080, −1.39375580494728651208113420178,
1.39375580494728651208113420178, 2.14898913496092963575229255080, 3.91210488051277779320076234638, 5.04471008961793371383618088791, 6.12115484502975746031593422200, 6.86414604779547030910547339232, 8.042238744681496704727000706394, 8.713426150404355454628361302705, 9.383700722197999637154978992730, 10.39359408099131626317482013831
