| L(s) = 1 | + 2·5-s + 2·7-s − 13-s − 2·17-s − 4·23-s − 25-s − 2·29-s + 2·31-s + 4·35-s − 6·37-s + 6·41-s − 2·43-s − 6·47-s − 3·49-s + 6·53-s − 8·59-s + 2·61-s − 2·65-s − 8·67-s + 2·71-s − 14·73-s + 4·79-s + 8·83-s − 4·85-s + 14·89-s − 2·91-s + 2·97-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.755·7-s − 0.277·13-s − 0.485·17-s − 0.834·23-s − 1/5·25-s − 0.371·29-s + 0.359·31-s + 0.676·35-s − 0.986·37-s + 0.937·41-s − 0.304·43-s − 0.875·47-s − 3/7·49-s + 0.824·53-s − 1.04·59-s + 0.256·61-s − 0.248·65-s − 0.977·67-s + 0.237·71-s − 1.63·73-s + 0.450·79-s + 0.878·83-s − 0.433·85-s + 1.48·89-s − 0.209·91-s + 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14976 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14976 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 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
−16.24750085136725, −15.92549565739594, −14.94122605268366, −14.80903030645507, −13.99605356257957, −13.63330361184887, −13.14728445574725, −12.35077406884110, −11.86656160712968, −11.25861916888874, −10.59775900472570, −10.13312682374587, −9.459844275609143, −8.989246833635842, −8.209993379337138, −7.743987639394050, −6.973932369502170, −6.279514149005588, −5.727746059223895, −5.057534119054296, −4.454296177918498, −3.667411511669169, −2.676103784883713, −1.979984948572915, −1.384453377726682, 0,
1.384453377726682, 1.979984948572915, 2.676103784883713, 3.667411511669169, 4.454296177918498, 5.057534119054296, 5.727746059223895, 6.279514149005588, 6.973932369502170, 7.743987639394050, 8.209993379337138, 8.989246833635842, 9.459844275609143, 10.13312682374587, 10.59775900472570, 11.25861916888874, 11.86656160712968, 12.35077406884110, 13.14728445574725, 13.63330361184887, 13.99605356257957, 14.80903030645507, 14.94122605268366, 15.92549565739594, 16.24750085136725
