| L(s) = 1 | − 2-s − 4-s − 5-s + 3·8-s + 10-s − 2·11-s − 13-s − 16-s − 8·17-s − 8·19-s + 20-s + 2·22-s + 25-s + 26-s + 4·29-s + 4·31-s − 5·32-s + 8·34-s + 4·37-s + 8·38-s − 3·40-s − 6·41-s − 10·43-s + 2·44-s + 6·47-s − 50-s + 52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.447·5-s + 1.06·8-s + 0.316·10-s − 0.603·11-s − 0.277·13-s − 1/4·16-s − 1.94·17-s − 1.83·19-s + 0.223·20-s + 0.426·22-s + 1/5·25-s + 0.196·26-s + 0.742·29-s + 0.718·31-s − 0.883·32-s + 1.37·34-s + 0.657·37-s + 1.29·38-s − 0.474·40-s − 0.937·41-s − 1.52·43-s + 0.301·44-s + 0.875·47-s − 0.141·50-s + 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28665 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28665 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 + 8 T + p T^{2} \) | 1.17.i |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−15.66078001317146, −15.26218055737868, −14.84890711850907, −14.04463172712036, −13.53599150306357, −13.02758448288467, −12.73402742902868, −11.88946286208969, −11.33057813834002, −10.69466261204451, −10.37166236053494, −9.829651417094885, −8.987431348740620, −8.658663010568642, −8.244042233052895, −7.686742472230416, −6.815821692505039, −6.585534736024653, −5.640594833304514, −4.690039286434039, −4.529192071405946, −3.912658681554736, −2.826647175916803, −2.210691297395300, −1.346917746025001, 0, 0,
1.346917746025001, 2.210691297395300, 2.826647175916803, 3.912658681554736, 4.529192071405946, 4.690039286434039, 5.640594833304514, 6.585534736024653, 6.815821692505039, 7.686742472230416, 8.244042233052895, 8.658663010568642, 8.987431348740620, 9.829651417094885, 10.37166236053494, 10.69466261204451, 11.33057813834002, 11.88946286208969, 12.73402742902868, 13.02758448288467, 13.53599150306357, 14.04463172712036, 14.84890711850907, 15.26218055737868, 15.66078001317146
