| L(s) = 1 | + 2-s + 4-s + 2·5-s + 2·7-s + 8-s + 2·10-s − 5·11-s + 2·14-s + 16-s − 6·17-s + 2·20-s − 5·22-s − 5·23-s − 25-s + 2·28-s + 4·29-s − 10·31-s + 32-s − 6·34-s + 4·35-s − 6·37-s + 2·40-s − 2·43-s − 5·44-s − 5·46-s + 13·47-s − 3·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.755·7-s + 0.353·8-s + 0.632·10-s − 1.50·11-s + 0.534·14-s + 1/4·16-s − 1.45·17-s + 0.447·20-s − 1.06·22-s − 1.04·23-s − 1/5·25-s + 0.377·28-s + 0.742·29-s − 1.79·31-s + 0.176·32-s − 1.02·34-s + 0.676·35-s − 0.986·37-s + 0.316·40-s − 0.304·43-s − 0.753·44-s − 0.737·46-s + 1.89·47-s − 3/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 5 T + p T^{2} \) | 1.23.f |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 - 13 T + p T^{2} \) | 1.47.an |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 5 T + p T^{2} \) | 1.59.af |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 + 3 T + p T^{2} \) | 1.83.d |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 17 T + p T^{2} \) | 1.97.r |
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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
−7.31694180420915091546569901948, −6.61851187253342831095058034696, −5.67460525191781674404261372892, −5.49277430360681206338021741730, −4.63941852762793200397421995221, −4.03765140602725412540585790000, −2.90811037187338101016756444042, −2.18628932775391499236563945626, −1.69998515585316327554617968183, 0,
1.69998515585316327554617968183, 2.18628932775391499236563945626, 2.90811037187338101016756444042, 4.03765140602725412540585790000, 4.63941852762793200397421995221, 5.49277430360681206338021741730, 5.67460525191781674404261372892, 6.61851187253342831095058034696, 7.31694180420915091546569901948
