| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 7-s − 8-s − 2·9-s − 12-s + 5·13-s − 14-s + 16-s − 3·17-s + 2·18-s − 21-s + 3·23-s + 24-s − 5·25-s − 5·26-s + 5·27-s + 28-s + 9·29-s + 4·31-s − 32-s + 3·34-s − 2·36-s − 2·37-s − 5·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s − 0.353·8-s − 2/3·9-s − 0.288·12-s + 1.38·13-s − 0.267·14-s + 1/4·16-s − 0.727·17-s + 0.471·18-s − 0.218·21-s + 0.625·23-s + 0.204·24-s − 25-s − 0.980·26-s + 0.962·27-s + 0.188·28-s + 1.67·29-s + 0.718·31-s − 0.176·32-s + 0.514·34-s − 1/3·36-s − 0.328·37-s − 0.800·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87362 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87362 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| 19 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 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
−14.02821811599199, −13.68193162493344, −13.23823873189857, −12.50546101228214, −11.93659214803958, −11.56436777054945, −11.18926970261941, −10.69452787954675, −10.30565059483203, −9.683595389693945, −9.005317940707388, −8.603938460289539, −8.186164189946683, −7.806242017998415, −6.768552899533336, −6.585148978114744, −6.094879536571834, −5.436227809919966, −4.912642241782150, −4.274636888869064, −3.496382189411865, −2.929536986401793, −2.219366648431818, −1.432082253544456, −0.8465226721890109, 0,
0.8465226721890109, 1.432082253544456, 2.219366648431818, 2.929536986401793, 3.496382189411865, 4.274636888869064, 4.912642241782150, 5.436227809919966, 6.094879536571834, 6.585148978114744, 6.768552899533336, 7.806242017998415, 8.186164189946683, 8.603938460289539, 9.005317940707388, 9.683595389693945, 10.30565059483203, 10.69452787954675, 11.18926970261941, 11.56436777054945, 11.93659214803958, 12.50546101228214, 13.23823873189857, 13.68193162493344, 14.02821811599199
