| L(s) = 1 | + 3-s − 5-s + 9-s + 4·11-s − 15-s − 6·17-s + 4·19-s + 25-s + 27-s − 2·29-s + 8·31-s + 4·33-s + 2·37-s + 6·41-s + 12·43-s − 45-s − 8·47-s − 7·49-s − 6·51-s + 6·53-s − 4·55-s + 4·57-s − 12·59-s + 14·61-s − 4·67-s − 8·71-s + 6·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s + 1.20·11-s − 0.258·15-s − 1.45·17-s + 0.917·19-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 1.43·31-s + 0.696·33-s + 0.328·37-s + 0.937·41-s + 1.82·43-s − 0.149·45-s − 1.16·47-s − 49-s − 0.840·51-s + 0.824·53-s − 0.539·55-s + 0.529·57-s − 1.56·59-s + 1.79·61-s − 0.488·67-s − 0.949·71-s + 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.816754900\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.816754900\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 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
−15.71290410576442, −15.06301411441402, −14.52449734798716, −14.09421576657612, −13.51950634324438, −12.98149378910381, −12.36735668737330, −11.72671896350033, −11.32266891237360, −10.78517296142287, −9.901910932414441, −9.443167059302883, −8.918721413578706, −8.421002657289783, −7.700909659845686, −7.201296759660302, −6.502446482574887, −6.065407036249992, −5.027009421317684, −4.333254017558968, −3.964291900138335, −3.109072229734202, −2.465768456185617, −1.545396641207692, −0.6974821817539960,
0.6974821817539960, 1.545396641207692, 2.465768456185617, 3.109072229734202, 3.964291900138335, 4.333254017558968, 5.027009421317684, 6.065407036249992, 6.502446482574887, 7.201296759660302, 7.700909659845686, 8.421002657289783, 8.918721413578706, 9.443167059302883, 9.901910932414441, 10.78517296142287, 11.32266891237360, 11.72671896350033, 12.36735668737330, 12.98149378910381, 13.51950634324438, 14.09421576657612, 14.52449734798716, 15.06301411441402, 15.71290410576442
