| L(s) = 1 | + 5-s − 7-s − 11-s − 13-s + 17-s − 5·19-s − 23-s − 4·25-s + 2·29-s + 6·31-s − 35-s + 8·37-s − 5·41-s + 43-s − 2·47-s + 49-s + 6·53-s − 55-s − 2·59-s + 8·61-s − 65-s + 4·67-s − 12·71-s + 14·73-s + 77-s + 10·79-s − 6·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.377·7-s − 0.301·11-s − 0.277·13-s + 0.242·17-s − 1.14·19-s − 0.208·23-s − 4/5·25-s + 0.371·29-s + 1.07·31-s − 0.169·35-s + 1.31·37-s − 0.780·41-s + 0.152·43-s − 0.291·47-s + 1/7·49-s + 0.824·53-s − 0.134·55-s − 0.260·59-s + 1.02·61-s − 0.124·65-s + 0.488·67-s − 1.42·71-s + 1.63·73-s + 0.113·77-s + 1.12·79-s − 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 - T \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−16.07589498519085, −15.66870450079915, −14.92473624287185, −14.65353346326275, −13.75188565298233, −13.48055212749206, −12.87984652215984, −12.28312535568583, −11.78220763294209, −11.07882207043824, −10.43112027266776, −9.937581040916741, −9.530789412839086, −8.738158590439801, −8.154646156757760, −7.641673323962410, −6.728586952838325, −6.342472053400689, −5.689216276058601, −4.989959514324694, −4.268826896228807, −3.605766038346684, −2.625039129306777, −2.196994986179635, −1.101366820089860, 0,
1.101366820089860, 2.196994986179635, 2.625039129306777, 3.605766038346684, 4.268826896228807, 4.989959514324694, 5.689216276058601, 6.342472053400689, 6.728586952838325, 7.641673323962410, 8.154646156757760, 8.738158590439801, 9.530789412839086, 9.937581040916741, 10.43112027266776, 11.07882207043824, 11.78220763294209, 12.28312535568583, 12.87984652215984, 13.48055212749206, 13.75188565298233, 14.65353346326275, 14.92473624287185, 15.66870450079915, 16.07589498519085
