| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s − 6·11-s + 12-s + 13-s + 16-s + 2·17-s + 18-s + 19-s − 6·22-s + 23-s + 24-s + 26-s + 27-s − 9·29-s + 2·31-s + 32-s − 6·33-s + 2·34-s + 36-s − 10·37-s + 38-s + 39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 1.80·11-s + 0.288·12-s + 0.277·13-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.229·19-s − 1.27·22-s + 0.208·23-s + 0.204·24-s + 0.196·26-s + 0.192·27-s − 1.67·29-s + 0.359·31-s + 0.176·32-s − 1.04·33-s + 0.342·34-s + 1/6·36-s − 1.64·37-s + 0.162·38-s + 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 139650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 19 | \( 1 - T \) | |
| good | 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 11 T + p T^{2} \) | 1.47.al |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 - 11 T + p T^{2} \) | 1.59.al |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 4 T + p T^{2} \) | 1.97.ae |
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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
−13.58024389506217, −13.21792107557376, −12.84991161641079, −12.33658766421013, −11.91848982241957, −11.16311115227145, −10.84219669939760, −10.26473321199860, −10.01445063258889, −9.224627547681698, −8.774454754656066, −8.219636394627857, −7.594721914659629, −7.430619151970423, −6.860814128704009, −6.029831601149519, −5.572001422146605, −5.197297363647609, −4.653777405357206, −3.883677251197299, −3.486092393698787, −2.895747747023268, −2.344125497474385, −1.851535319075759, −0.9570576089161345, 0,
0.9570576089161345, 1.851535319075759, 2.344125497474385, 2.895747747023268, 3.486092393698787, 3.883677251197299, 4.653777405357206, 5.197297363647609, 5.572001422146605, 6.029831601149519, 6.860814128704009, 7.430619151970423, 7.594721914659629, 8.219636394627857, 8.774454754656066, 9.224627547681698, 10.01445063258889, 10.26473321199860, 10.84219669939760, 11.16311115227145, 11.91848982241957, 12.33658766421013, 12.84991161641079, 13.21792107557376, 13.58024389506217
