| L(s) = 1 | + 2-s − 3·3-s + 4-s − 3·6-s + 8-s + 6·9-s − 6·11-s − 3·12-s − 4·13-s + 16-s + 6·18-s − 6·22-s + 4·23-s − 3·24-s − 4·26-s − 9·27-s + 6·29-s + 7·31-s + 32-s + 18·33-s + 6·36-s + 9·37-s + 12·39-s − 43-s − 6·44-s + 4·46-s − 6·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.73·3-s + 1/2·4-s − 1.22·6-s + 0.353·8-s + 2·9-s − 1.80·11-s − 0.866·12-s − 1.10·13-s + 1/4·16-s + 1.41·18-s − 1.27·22-s + 0.834·23-s − 0.612·24-s − 0.784·26-s − 1.73·27-s + 1.11·29-s + 1.25·31-s + 0.176·32-s + 3.13·33-s + 36-s + 1.47·37-s + 1.92·39-s − 0.152·43-s − 0.904·44-s + 0.589·46-s − 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 193550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 79 | \( 1 + T \) | |
| good | 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 - 9 T + p T^{2} \) | 1.37.aj |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−13.12956878826331, −12.79453654124633, −12.34596824653495, −12.03202047020958, −11.45012866304703, −11.09055472849030, −10.63976639090314, −10.16205963627693, −9.918817577480021, −9.306078884347050, −8.297893787147313, −7.937268709598996, −7.419615017975602, −6.869184625861738, −6.491745699387706, −5.928161228311178, −5.438528473876567, −5.032620147028643, −4.612409560552157, −4.399304280474814, −3.320880823533649, −2.698286764121584, −2.361988973380329, −1.347920622118070, −0.6885919980420858, 0,
0.6885919980420858, 1.347920622118070, 2.361988973380329, 2.698286764121584, 3.320880823533649, 4.399304280474814, 4.612409560552157, 5.032620147028643, 5.438528473876567, 5.928161228311178, 6.491745699387706, 6.869184625861738, 7.419615017975602, 7.937268709598996, 8.297893787147313, 9.306078884347050, 9.918817577480021, 10.16205963627693, 10.63976639090314, 11.09055472849030, 11.45012866304703, 12.03202047020958, 12.34596824653495, 12.79453654124633, 13.12956878826331
