| L(s) = 1 | − 5-s + 7-s − 2·13-s − 2·17-s − 4·19-s + 25-s + 2·29-s + 4·31-s − 35-s − 2·37-s + 2·41-s − 8·43-s + 4·47-s + 49-s − 2·53-s + 12·59-s + 14·61-s + 2·65-s − 8·67-s + 8·71-s − 2·73-s + 8·79-s − 4·83-s + 2·85-s + 18·89-s − 2·91-s + 4·95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s − 0.554·13-s − 0.485·17-s − 0.917·19-s + 1/5·25-s + 0.371·29-s + 0.718·31-s − 0.169·35-s − 0.328·37-s + 0.312·41-s − 1.21·43-s + 0.583·47-s + 1/7·49-s − 0.274·53-s + 1.56·59-s + 1.79·61-s + 0.248·65-s − 0.977·67-s + 0.949·71-s − 0.234·73-s + 0.900·79-s − 0.439·83-s + 0.216·85-s + 1.90·89-s − 0.209·91-s + 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−15.85960233393140, −15.36457639590078, −14.79812501603846, −14.46808864295425, −13.71165453316843, −13.21070007336571, −12.59677397050399, −12.07290705583705, −11.50535191582248, −11.07981178848819, −10.26381126071857, −10.03367328728605, −9.048916715684431, −8.643568651708998, −8.034512584087757, −7.505792714830146, −6.702837260722253, −6.399481863014592, −5.354185768809092, −4.898087783459106, −4.187818426872092, −3.630151936581736, −2.629290565444495, −2.097959899160967, −1.021497793605958, 0,
1.021497793605958, 2.097959899160967, 2.629290565444495, 3.630151936581736, 4.187818426872092, 4.898087783459106, 5.354185768809092, 6.399481863014592, 6.702837260722253, 7.505792714830146, 8.034512584087757, 8.643568651708998, 9.048916715684431, 10.03367328728605, 10.26381126071857, 11.07981178848819, 11.50535191582248, 12.07290705583705, 12.59677397050399, 13.21070007336571, 13.71165453316843, 14.46808864295425, 14.79812501603846, 15.36457639590078, 15.85960233393140
