| L(s) = 1 | + 2-s + 3-s + 4-s − 3·5-s + 6-s + 8-s + 9-s − 3·10-s + 5·11-s + 12-s − 3·15-s + 16-s + 18-s − 8·19-s − 3·20-s + 5·22-s − 2·23-s + 24-s + 4·25-s + 27-s − 9·29-s − 3·30-s + 5·31-s + 32-s + 5·33-s + 36-s − 6·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.948·10-s + 1.50·11-s + 0.288·12-s − 0.774·15-s + 1/4·16-s + 0.235·18-s − 1.83·19-s − 0.670·20-s + 1.06·22-s − 0.417·23-s + 0.204·24-s + 4/5·25-s + 0.192·27-s − 1.67·29-s − 0.547·30-s + 0.898·31-s + 0.176·32-s + 0.870·33-s + 1/6·36-s − 0.986·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 293046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 293046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + T + p T^{2} \) | 1.73.b |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 16 T + p T^{2} \) | 1.89.aq |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
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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
−12.88774380927530, −12.38536289954222, −12.07263231408144, −11.76183154982914, −11.15309316600658, −10.80098857588855, −10.39533627237860, −9.500880056623128, −9.318892097815854, −8.659914261483106, −8.135707284221224, −8.003363118804756, −7.213821935226140, −6.893576915014313, −6.366976894494267, −6.044851034319214, −5.096118203897912, −4.735500442126232, −4.014712356422663, −3.742409027646487, −3.674737903944031, −2.765245210986517, −2.076700042895649, −1.687144346518683, −0.7987605990850253, 0,
0.7987605990850253, 1.687144346518683, 2.076700042895649, 2.765245210986517, 3.674737903944031, 3.742409027646487, 4.014712356422663, 4.735500442126232, 5.096118203897912, 6.044851034319214, 6.366976894494267, 6.893576915014313, 7.213821935226140, 8.003363118804756, 8.135707284221224, 8.659914261483106, 9.318892097815854, 9.500880056623128, 10.39533627237860, 10.80098857588855, 11.15309316600658, 11.76183154982914, 12.07263231408144, 12.38536289954222, 12.88774380927530
