| L(s) = 1 | + 2·2-s + 3-s + 2·4-s + 5-s + 2·6-s − 2·9-s + 2·10-s − 11-s + 2·12-s + 15-s − 4·16-s + 2·17-s − 4·18-s + 2·20-s − 2·22-s − 23-s − 4·25-s − 5·27-s + 2·30-s + 7·31-s − 8·32-s − 33-s + 4·34-s − 4·36-s − 3·37-s − 8·41-s − 6·43-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s + 0.447·5-s + 0.816·6-s − 2/3·9-s + 0.632·10-s − 0.301·11-s + 0.577·12-s + 0.258·15-s − 16-s + 0.485·17-s − 0.942·18-s + 0.447·20-s − 0.426·22-s − 0.208·23-s − 4/5·25-s − 0.962·27-s + 0.365·30-s + 1.25·31-s − 1.41·32-s − 0.174·33-s + 0.685·34-s − 2/3·36-s − 0.493·37-s − 1.24·41-s − 0.914·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91091 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91091 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.893617085\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.893617085\) |
| \(L(\frac{3}{2})\) |
|
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 | 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 2 | \( 1 - p T + p T^{2} \) | 1.2.ac |
| 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 5 T + p T^{2} \) | 1.59.af |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−13.85978003440859, −13.43687378582268, −13.18024385194583, −12.33087077611465, −12.13245052606808, −11.56302424789600, −11.12225507336051, −10.40068831660324, −9.863555638896463, −9.470673834525833, −8.634743349311128, −8.483159430848148, −7.720193241531513, −7.207168194054284, −6.381679396186418, −6.138106138254212, −5.490975622415585, −5.112555946617895, −4.500979435636870, −3.844396521775447, −3.261551978760056, −2.925590858141073, −2.197786987136438, −1.724949377882166, −0.4893732582597292,
0.4893732582597292, 1.724949377882166, 2.197786987136438, 2.925590858141073, 3.261551978760056, 3.844396521775447, 4.500979435636870, 5.112555946617895, 5.490975622415585, 6.138106138254212, 6.381679396186418, 7.207168194054284, 7.720193241531513, 8.483159430848148, 8.634743349311128, 9.470673834525833, 9.863555638896463, 10.40068831660324, 11.12225507336051, 11.56302424789600, 12.13245052606808, 12.33087077611465, 13.18024385194583, 13.43687378582268, 13.85978003440859
