| L(s) = 1 | + 2-s + 4-s − 3·5-s + 8-s − 3·10-s − 3·11-s + 4·13-s + 16-s − 4·17-s + 2·19-s − 3·20-s − 3·22-s + 6·23-s + 4·25-s + 4·26-s − 5·29-s + 7·31-s + 32-s − 4·34-s + 37-s + 2·38-s − 3·40-s − 8·41-s − 8·43-s − 3·44-s + 6·46-s + 10·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.34·5-s + 0.353·8-s − 0.948·10-s − 0.904·11-s + 1.10·13-s + 1/4·16-s − 0.970·17-s + 0.458·19-s − 0.670·20-s − 0.639·22-s + 1.25·23-s + 4/5·25-s + 0.784·26-s − 0.928·29-s + 1.25·31-s + 0.176·32-s − 0.685·34-s + 0.164·37-s + 0.324·38-s − 0.474·40-s − 1.24·41-s − 1.21·43-s − 0.452·44-s + 0.884·46-s + 1.45·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32634 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32634 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.166698152\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.166698152\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 37 | \( 1 - T \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 + 11 T + p T^{2} \) | 1.53.l |
| 59 | \( 1 + T + p T^{2} \) | 1.59.b |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 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.ag |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 + 15 T + p T^{2} \) | 1.83.p |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
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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
−15.25714254732900, −14.61897039028576, −13.72790198934016, −13.56525764686233, −12.87491223714478, −12.54319635786718, −11.72275198128227, −11.41563132402711, −10.96822252487660, −10.51747386313019, −9.726055448078828, −8.937390166653022, −8.396517435945757, −7.913192113684944, −7.406925813664436, −6.666637666667698, −6.354639110637175, −5.242371637721389, −5.062096848624213, −4.206826042194454, −3.722257754253267, −3.141073827306937, −2.492535676845313, −1.467839056668096, −0.5037850538273112,
0.5037850538273112, 1.467839056668096, 2.492535676845313, 3.141073827306937, 3.722257754253267, 4.206826042194454, 5.062096848624213, 5.242371637721389, 6.354639110637175, 6.666637666667698, 7.406925813664436, 7.913192113684944, 8.396517435945757, 8.937390166653022, 9.726055448078828, 10.51747386313019, 10.96822252487660, 11.41563132402711, 11.72275198128227, 12.54319635786718, 12.87491223714478, 13.56525764686233, 13.72790198934016, 14.61897039028576, 15.25714254732900
