| L(s) = 1 | − 5-s + 5·7-s + 2·11-s + 3·17-s − 2·19-s + 4·23-s − 4·25-s + 6·29-s − 4·31-s − 5·35-s − 11·37-s + 8·41-s + 43-s − 9·47-s + 18·49-s + 12·53-s − 2·55-s − 6·59-s + 6·67-s − 7·71-s + 2·73-s + 10·77-s − 12·79-s + 16·83-s − 3·85-s − 10·89-s + 2·95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.88·7-s + 0.603·11-s + 0.727·17-s − 0.458·19-s + 0.834·23-s − 4/5·25-s + 1.11·29-s − 0.718·31-s − 0.845·35-s − 1.80·37-s + 1.24·41-s + 0.152·43-s − 1.31·47-s + 18/7·49-s + 1.64·53-s − 0.269·55-s − 0.781·59-s + 0.733·67-s − 0.830·71-s + 0.234·73-s + 1.13·77-s − 1.35·79-s + 1.75·83-s − 0.325·85-s − 1.05·89-s + 0.205·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.032471485\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.032471485\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 - 5 T + p T^{2} \) | 1.7.af |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 11 T + p T^{2} \) | 1.37.l |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 6 T + p T^{2} \) | 1.67.ag |
| 71 | \( 1 + 7 T + p T^{2} \) | 1.71.h |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−15.26410098622960, −14.86778649972855, −14.21642120043307, −14.13697837391356, −13.32656264949278, −12.53365440611421, −12.07955195610263, −11.54100284222820, −11.22160598123932, −10.54497234478344, −10.10391477186527, −9.149687234626826, −8.674935247302292, −8.230564444110898, −7.566799262608687, −7.220404520059289, −6.388023957890457, −5.582892009350191, −5.071490293022071, −4.460380583031664, −3.908061552387036, −3.139609466000448, −2.115392447164319, −1.533463758274887, −0.7371735128587570,
0.7371735128587570, 1.533463758274887, 2.115392447164319, 3.139609466000448, 3.908061552387036, 4.460380583031664, 5.071490293022071, 5.582892009350191, 6.388023957890457, 7.220404520059289, 7.566799262608687, 8.230564444110898, 8.674935247302292, 9.149687234626826, 10.10391477186527, 10.54497234478344, 11.22160598123932, 11.54100284222820, 12.07955195610263, 12.53365440611421, 13.32656264949278, 14.13697837391356, 14.21642120043307, 14.86778649972855, 15.26410098622960
