| L(s) = 1 | − 2-s + 4-s − 8-s + 4·13-s + 16-s + 6·17-s − 2·19-s + 3·23-s − 5·25-s − 4·26-s + 6·29-s − 5·31-s − 32-s − 6·34-s + 8·37-s + 2·38-s + 3·41-s + 2·43-s − 3·46-s − 3·47-s + 5·50-s + 4·52-s − 6·53-s − 6·58-s + 12·59-s − 8·61-s + 5·62-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 1.10·13-s + 1/4·16-s + 1.45·17-s − 0.458·19-s + 0.625·23-s − 25-s − 0.784·26-s + 1.11·29-s − 0.898·31-s − 0.176·32-s − 1.02·34-s + 1.31·37-s + 0.324·38-s + 0.468·41-s + 0.304·43-s − 0.442·46-s − 0.437·47-s + 0.707·50-s + 0.554·52-s − 0.824·53-s − 0.787·58-s + 1.56·59-s − 1.02·61-s + 0.635·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.637906200\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.637906200\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 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
−8.069642655004683675584952567559, −7.27611978121869643220459621962, −6.47774436693225369667869317565, −5.91126990195501981664003203735, −5.21766716670261490424785581295, −4.14057007448049792925637458303, −3.44181280440197341091078848814, −2.60462880871233570917807884019, −1.54811197421885365449497759949, −0.75499762600709429772212805596,
0.75499762600709429772212805596, 1.54811197421885365449497759949, 2.60462880871233570917807884019, 3.44181280440197341091078848814, 4.14057007448049792925637458303, 5.21766716670261490424785581295, 5.91126990195501981664003203735, 6.47774436693225369667869317565, 7.27611978121869643220459621962, 8.069642655004683675584952567559
