| L(s) = 1 | − 5-s − 4·7-s + 13-s + 2·17-s + 4·19-s + 8·23-s + 25-s + 2·29-s + 8·31-s + 4·35-s − 2·37-s + 6·41-s + 12·43-s + 9·49-s + 10·53-s + 10·61-s − 65-s − 4·67-s − 16·71-s − 6·73-s + 8·79-s + 4·83-s − 2·85-s + 14·89-s − 4·91-s − 4·95-s − 6·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.51·7-s + 0.277·13-s + 0.485·17-s + 0.917·19-s + 1.66·23-s + 1/5·25-s + 0.371·29-s + 1.43·31-s + 0.676·35-s − 0.328·37-s + 0.937·41-s + 1.82·43-s + 9/7·49-s + 1.37·53-s + 1.28·61-s − 0.124·65-s − 0.488·67-s − 1.89·71-s − 0.702·73-s + 0.900·79-s + 0.439·83-s − 0.216·85-s + 1.48·89-s − 0.419·91-s − 0.410·95-s − 0.609·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.140448484\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.140448484\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−14.93012778181735, −14.34773497979992, −13.72555115467002, −13.23021847845060, −12.84933552940351, −12.22880768956976, −11.83412301563147, −11.22915801113040, −10.51362086622541, −10.15340527666171, −9.525656183230270, −8.987696065383927, −8.632729738141823, −7.632163431220872, −7.367714479813899, −6.694099894394824, −6.168350575892231, −5.590378522573793, −4.886622838582645, −4.151309219019836, −3.510406589321264, −2.956557869482777, −2.511982340116743, −1.078906588894156, −0.6614765689140241,
0.6614765689140241, 1.078906588894156, 2.511982340116743, 2.956557869482777, 3.510406589321264, 4.151309219019836, 4.886622838582645, 5.590378522573793, 6.168350575892231, 6.694099894394824, 7.367714479813899, 7.632163431220872, 8.632729738141823, 8.987696065383927, 9.525656183230270, 10.15340527666171, 10.51362086622541, 11.22915801113040, 11.83412301563147, 12.22880768956976, 12.84933552940351, 13.23021847845060, 13.72555115467002, 14.34773497979992, 14.93012778181735
