| L(s) = 1 | + 2·7-s − 11-s − 4·13-s − 4·19-s + 4·23-s + 2·29-s − 8·31-s + 2·37-s + 6·41-s + 6·43-s − 4·47-s − 3·49-s + 6·53-s + 4·59-s − 6·61-s + 16·71-s + 4·73-s − 2·77-s − 8·79-s + 6·83-s − 14·89-s − 8·91-s − 6·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 0.301·11-s − 1.10·13-s − 0.917·19-s + 0.834·23-s + 0.371·29-s − 1.43·31-s + 0.328·37-s + 0.937·41-s + 0.914·43-s − 0.583·47-s − 3/7·49-s + 0.824·53-s + 0.520·59-s − 0.768·61-s + 1.89·71-s + 0.468·73-s − 0.227·77-s − 0.900·79-s + 0.658·83-s − 1.48·89-s − 0.838·91-s − 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−15.89860805547676, −15.30018631438142, −14.74938000010171, −14.48794895250278, −13.87738947774096, −13.10381927765663, −12.60916413160209, −12.28795377815723, −11.29537695987142, −11.13755422131962, −10.46870950191267, −9.814934062023056, −9.233444954426031, −8.636856988369024, −8.009155041721153, −7.441182102712207, −6.960816291197428, −6.159724423850516, −5.425483581664314, −4.890528131044805, −4.338823104628932, −3.540915123756664, −2.571065779145302, −2.114731425546232, −1.103193440697803, 0,
1.103193440697803, 2.114731425546232, 2.571065779145302, 3.540915123756664, 4.338823104628932, 4.890528131044805, 5.425483581664314, 6.159724423850516, 6.960816291197428, 7.441182102712207, 8.009155041721153, 8.636856988369024, 9.233444954426031, 9.814934062023056, 10.46870950191267, 11.13755422131962, 11.29537695987142, 12.28795377815723, 12.60916413160209, 13.10381927765663, 13.87738947774096, 14.48794895250278, 14.74938000010171, 15.30018631438142, 15.89860805547676
