| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 2·7-s − 8-s − 2·9-s + 12-s + 13-s − 2·14-s + 16-s + 17-s + 2·18-s + 2·19-s + 2·21-s + 5·23-s − 24-s − 26-s − 5·27-s + 2·28-s + 29-s − 6·31-s − 32-s − 34-s − 2·36-s + 2·37-s − 2·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.755·7-s − 0.353·8-s − 2/3·9-s + 0.288·12-s + 0.277·13-s − 0.534·14-s + 1/4·16-s + 0.242·17-s + 0.471·18-s + 0.458·19-s + 0.436·21-s + 1.04·23-s − 0.204·24-s − 0.196·26-s − 0.962·27-s + 0.377·28-s + 0.185·29-s − 1.07·31-s − 0.176·32-s − 0.171·34-s − 1/3·36-s + 0.328·37-s − 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 13 T + p T^{2} \) | 1.53.an |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 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
−14.38875704347992, −13.85699434433777, −13.33788824955281, −12.78808018819971, −12.17141049454755, −11.58051298797279, −11.20165140798694, −10.89450182330524, −10.13730707565963, −9.683684256500042, −9.062711039609837, −8.727952408111927, −8.224182464351450, −7.805602241420012, −7.246943064935545, −6.725792570816274, −6.026728991726253, −5.334846081839746, −5.070396674931781, −4.102038562203424, −3.484648008703256, −2.929107709928102, −2.312898736542247, −1.613528916153506, −0.9935795074284654, 0,
0.9935795074284654, 1.613528916153506, 2.312898736542247, 2.929107709928102, 3.484648008703256, 4.102038562203424, 5.070396674931781, 5.334846081839746, 6.026728991726253, 6.725792570816274, 7.246943064935545, 7.805602241420012, 8.224182464351450, 8.727952408111927, 9.062711039609837, 9.683684256500042, 10.13730707565963, 10.89450182330524, 11.20165140798694, 11.58051298797279, 12.17141049454755, 12.78808018819971, 13.33788824955281, 13.85699434433777, 14.38875704347992
