| L(s) = 1 | − 2-s + 3·3-s + 4-s − 3·5-s − 3·6-s + 3·7-s − 8-s + 6·9-s + 3·10-s + 3·12-s − 13-s − 3·14-s − 9·15-s + 16-s + 5·17-s − 6·18-s + 19-s − 3·20-s + 9·21-s + 6·23-s − 3·24-s + 4·25-s + 26-s + 9·27-s + 3·28-s − 8·29-s + 9·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.73·3-s + 1/2·4-s − 1.34·5-s − 1.22·6-s + 1.13·7-s − 0.353·8-s + 2·9-s + 0.948·10-s + 0.866·12-s − 0.277·13-s − 0.801·14-s − 2.32·15-s + 1/4·16-s + 1.21·17-s − 1.41·18-s + 0.229·19-s − 0.670·20-s + 1.96·21-s + 1.25·23-s − 0.612·24-s + 4/5·25-s + 0.196·26-s + 1.73·27-s + 0.566·28-s − 1.48·29-s + 1.64·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 494 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 494 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.706963464\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.706963464\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| 19 | \( 1 - T \) | |
| good | 3 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 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
−10.86543252343371964580538640362, −9.795570901256845188633743267131, −8.923699200788337239745949827213, −8.117039731648527262775096054287, −7.78451188128530689930062166224, −7.06464986310461074913571294057, −4.99190280565554964852691201008, −3.80333294491356206353212848867, −2.92557868251333773162660053529, −1.47455662771436422559639220199,
1.47455662771436422559639220199, 2.92557868251333773162660053529, 3.80333294491356206353212848867, 4.99190280565554964852691201008, 7.06464986310461074913571294057, 7.78451188128530689930062166224, 8.117039731648527262775096054287, 8.923699200788337239745949827213, 9.795570901256845188633743267131, 10.86543252343371964580538640362
