| L(s) = 1 | + 2-s + 4-s + 3·5-s + 8-s + 3·10-s + 4·13-s + 16-s + 6·17-s − 5·19-s + 3·20-s + 4·25-s + 4·26-s + 3·29-s + 4·31-s + 32-s + 6·34-s − 10·37-s − 5·38-s + 3·40-s − 12·41-s − 43-s + 12·47-s + 4·50-s + 4·52-s − 6·53-s + 3·58-s + 6·59-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.34·5-s + 0.353·8-s + 0.948·10-s + 1.10·13-s + 1/4·16-s + 1.45·17-s − 1.14·19-s + 0.670·20-s + 4/5·25-s + 0.784·26-s + 0.557·29-s + 0.718·31-s + 0.176·32-s + 1.02·34-s − 1.64·37-s − 0.811·38-s + 0.474·40-s − 1.87·41-s − 0.152·43-s + 1.75·47-s + 0.565·50-s + 0.554·52-s − 0.824·53-s + 0.393·58-s + 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320166 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.122523758\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.122523758\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 + 5 T + p T^{2} \) | 1.73.f |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
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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
−12.60792925056436, −12.33664010211561, −11.85118436135987, −11.23441758636850, −10.81764812003842, −10.26258477316245, −10.00414809430663, −9.686184935151427, −8.798353742006794, −8.450912390733496, −8.259134438671640, −7.252656760659002, −6.957372827817847, −6.400366769805874, −5.938193138259900, −5.672489435838733, −5.122164398417892, −4.674210271988988, −3.933833040698806, −3.467844561591461, −3.035213855309963, −2.248846314898595, −1.883372315487913, −1.295048894654815, −0.6585891761547498,
0.6585891761547498, 1.295048894654815, 1.883372315487913, 2.248846314898595, 3.035213855309963, 3.467844561591461, 3.933833040698806, 4.674210271988988, 5.122164398417892, 5.672489435838733, 5.938193138259900, 6.400366769805874, 6.957372827817847, 7.252656760659002, 8.259134438671640, 8.450912390733496, 8.798353742006794, 9.686184935151427, 10.00414809430663, 10.26258477316245, 10.81764812003842, 11.23441758636850, 11.85118436135987, 12.33664010211561, 12.60792925056436
