| L(s) = 1 | + 2-s + 4-s − 5-s + 4·7-s + 8-s − 10-s − 11-s + 4·14-s + 16-s − 2·19-s − 20-s − 22-s − 6·23-s + 25-s + 4·28-s − 6·29-s − 2·31-s + 32-s − 4·35-s + 4·37-s − 2·38-s − 40-s − 6·41-s + 8·43-s − 44-s − 6·46-s + 9·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.51·7-s + 0.353·8-s − 0.316·10-s − 0.301·11-s + 1.06·14-s + 1/4·16-s − 0.458·19-s − 0.223·20-s − 0.213·22-s − 1.25·23-s + 1/5·25-s + 0.755·28-s − 1.11·29-s − 0.359·31-s + 0.176·32-s − 0.676·35-s + 0.657·37-s − 0.324·38-s − 0.158·40-s − 0.937·41-s + 1.21·43-s − 0.150·44-s − 0.884·46-s + 9/7·49-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 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−13.52151362461457, −13.05514405868259, −12.44015552999254, −11.97761486998753, −11.75232142731170, −11.10378152970906, −10.69046175246397, −10.55654016372163, −9.572684357865708, −9.228735358814341, −8.450114253064446, −8.013290683293634, −7.767742561068465, −7.243668862151436, −6.630167559194929, −5.904926278450204, −5.631404978646832, −4.914120378647121, −4.575229853168814, −4.042953381717302, −3.578086965534630, −2.831304866898571, −2.028708073638160, −1.845945418454854, −0.9463576664042222, 0,
0.9463576664042222, 1.845945418454854, 2.028708073638160, 2.831304866898571, 3.578086965534630, 4.042953381717302, 4.575229853168814, 4.914120378647121, 5.631404978646832, 5.904926278450204, 6.630167559194929, 7.243668862151436, 7.767742561068465, 8.013290683293634, 8.450114253064446, 9.228735358814341, 9.572684357865708, 10.55654016372163, 10.69046175246397, 11.10378152970906, 11.75232142731170, 11.97761486998753, 12.44015552999254, 13.05514405868259, 13.52151362461457
