| L(s) = 1 | + 2-s − 2·3-s − 4-s − 2·6-s − 3·8-s + 9-s − 2·11-s + 2·12-s − 16-s + 18-s + 6·19-s − 2·22-s + 6·23-s + 6·24-s + 4·27-s − 6·29-s + 6·31-s + 5·32-s + 4·33-s − 36-s + 6·37-s + 6·38-s − 8·41-s − 6·43-s + 2·44-s + 6·46-s − 8·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s − 1/2·4-s − 0.816·6-s − 1.06·8-s + 1/3·9-s − 0.603·11-s + 0.577·12-s − 1/4·16-s + 0.235·18-s + 1.37·19-s − 0.426·22-s + 1.25·23-s + 1.22·24-s + 0.769·27-s − 1.11·29-s + 1.07·31-s + 0.883·32-s + 0.696·33-s − 1/6·36-s + 0.986·37-s + 0.973·38-s − 1.24·41-s − 0.914·43-s + 0.301·44-s + 0.884·46-s − 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−8.015679625846962591559675712205, −7.04600074781313758640204967260, −6.35999446976652482051560318045, −5.49783119119489739541857071135, −5.19068583229192921467737965262, −4.55507775314440764576963954263, −3.46796694228874079626916651898, −2.77631061358590955837322830796, −1.12085671907922116731672488683, 0,
1.12085671907922116731672488683, 2.77631061358590955837322830796, 3.46796694228874079626916651898, 4.55507775314440764576963954263, 5.19068583229192921467737965262, 5.49783119119489739541857071135, 6.35999446976652482051560318045, 7.04600074781313758640204967260, 8.015679625846962591559675712205
