| L(s) = 1 | − 2-s + 4-s + 5-s + 2·7-s − 8-s − 10-s + 11-s − 2·14-s + 16-s + 20-s − 22-s − 6·23-s + 25-s + 2·28-s − 6·29-s + 8·31-s − 32-s + 2·35-s − 2·37-s − 40-s + 10·41-s + 4·43-s + 44-s + 6·46-s + 8·47-s − 3·49-s − 50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.755·7-s − 0.353·8-s − 0.316·10-s + 0.301·11-s − 0.534·14-s + 1/4·16-s + 0.223·20-s − 0.213·22-s − 1.25·23-s + 1/5·25-s + 0.377·28-s − 1.11·29-s + 1.43·31-s − 0.176·32-s + 0.338·35-s − 0.328·37-s − 0.158·40-s + 1.56·41-s + 0.609·43-s + 0.150·44-s + 0.884·46-s + 1.16·47-s − 3/7·49-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 167310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 167310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.51481903401850, −12.99011352505925, −12.30500313497676, −12.08493113789706, −11.45779942766647, −11.05438384765050, −10.61323662154263, −10.10263639188782, −9.631386120124287, −9.161895283841300, −8.773452590352733, −8.108031049034235, −7.736565055255090, −7.416426124452790, −6.516476404067337, −6.313068156498190, −5.645939211608292, −5.219541819776175, −4.441295583240203, −4.038007907636185, −3.338654053735739, −2.461445053067873, −2.205870319298185, −1.429144054555850, −0.9378626661451819, 0,
0.9378626661451819, 1.429144054555850, 2.205870319298185, 2.461445053067873, 3.338654053735739, 4.038007907636185, 4.441295583240203, 5.219541819776175, 5.645939211608292, 6.313068156498190, 6.516476404067337, 7.416426124452790, 7.736565055255090, 8.108031049034235, 8.773452590352733, 9.161895283841300, 9.631386120124287, 10.10263639188782, 10.61323662154263, 11.05438384765050, 11.45779942766647, 12.08493113789706, 12.30500313497676, 12.99011352505925, 13.51481903401850
