| L(s) = 1 | − 2-s + 3·3-s + 4-s + 2·5-s − 3·6-s − 7-s − 8-s + 6·9-s − 2·10-s − 4·11-s + 3·12-s − 2·13-s + 14-s + 6·15-s + 16-s − 3·17-s − 6·18-s + 8·19-s + 2·20-s − 3·21-s + 4·22-s + 2·23-s − 3·24-s − 25-s + 2·26-s + 9·27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.73·3-s + 1/2·4-s + 0.894·5-s − 1.22·6-s − 0.377·7-s − 0.353·8-s + 2·9-s − 0.632·10-s − 1.20·11-s + 0.866·12-s − 0.554·13-s + 0.267·14-s + 1.54·15-s + 1/4·16-s − 0.727·17-s − 1.41·18-s + 1.83·19-s + 0.447·20-s − 0.654·21-s + 0.852·22-s + 0.417·23-s − 0.612·24-s − 1/5·25-s + 0.392·26-s + 1.73·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39326 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39326 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 53 | \( 1 \) | |
| good | 3 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 + 3 T + p T^{2} \) | 1.43.d |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 15 T + p T^{2} \) | 1.73.p |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 7 T + p T^{2} \) | 1.83.h |
| 89 | \( 1 + T + p T^{2} \) | 1.89.b |
| 97 | \( 1 - 13 T + p T^{2} \) | 1.97.an |
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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
−15.24320350460845, −14.29807005927802, −14.15429066485722, −13.51823939816205, −13.17453036901540, −12.73296875855385, −11.90178208082779, −11.40631652883325, −10.30346363465418, −10.09898283556961, −9.901140490636395, −9.034716398772769, −8.874814226358763, −8.205216740156961, −7.651080884874401, −7.179684577477313, −6.710020570759222, −5.834681882746691, −5.124421922327045, −4.601275692866996, −3.387924931067184, −3.121271747123918, −2.520098279608035, −1.927310250604382, −1.290259481858575, 0,
1.290259481858575, 1.927310250604382, 2.520098279608035, 3.121271747123918, 3.387924931067184, 4.601275692866996, 5.124421922327045, 5.834681882746691, 6.710020570759222, 7.179684577477313, 7.651080884874401, 8.205216740156961, 8.874814226358763, 9.034716398772769, 9.901140490636395, 10.09898283556961, 10.30346363465418, 11.40631652883325, 11.90178208082779, 12.73296875855385, 13.17453036901540, 13.51823939816205, 14.15429066485722, 14.29807005927802, 15.24320350460845
