| L(s) = 1 | + 2-s + 4-s + 3·5-s − 2·7-s + 8-s + 3·10-s − 6·11-s − 13-s − 2·14-s + 16-s + 5·19-s + 3·20-s − 6·22-s − 6·23-s + 4·25-s − 26-s − 2·28-s + 31-s + 32-s − 6·35-s − 2·37-s + 5·38-s + 3·40-s − 41-s + 8·43-s − 6·44-s − 6·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.34·5-s − 0.755·7-s + 0.353·8-s + 0.948·10-s − 1.80·11-s − 0.277·13-s − 0.534·14-s + 1/4·16-s + 1.14·19-s + 0.670·20-s − 1.27·22-s − 1.25·23-s + 4/5·25-s − 0.196·26-s − 0.377·28-s + 0.179·31-s + 0.176·32-s − 1.01·35-s − 0.328·37-s + 0.811·38-s + 0.474·40-s − 0.156·41-s + 1.21·43-s − 0.904·44-s − 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 213282 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 213282 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 17 | \( 1 \) | |
| 41 | \( 1 + T \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 3 T + p T^{2} \) | 1.83.d |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−13.38708174699003, −12.76439964205008, −12.46274777497813, −12.10365506783910, −11.34976961829566, −10.80707769608667, −10.40473636069564, −10.00495798941815, −9.510672571912640, −9.361487627926008, −8.294971151390403, −8.072132911196718, −7.371253322816288, −6.959151903378127, −6.396940021873336, −5.767833820050326, −5.503257463634836, −5.261097096451434, −4.487713186057782, −3.878169193517996, −3.216050802641507, −2.553111165583232, −2.460786067954149, −1.728818231063746, −0.8781454315988784, 0,
0.8781454315988784, 1.728818231063746, 2.460786067954149, 2.553111165583232, 3.216050802641507, 3.878169193517996, 4.487713186057782, 5.261097096451434, 5.503257463634836, 5.767833820050326, 6.396940021873336, 6.959151903378127, 7.371253322816288, 8.072132911196718, 8.294971151390403, 9.361487627926008, 9.510672571912640, 10.00495798941815, 10.40473636069564, 10.80707769608667, 11.34976961829566, 12.10365506783910, 12.46274777497813, 12.76439964205008, 13.38708174699003
