| L(s) = 1 | − 2·5-s − 7-s − 6·11-s + 2·13-s + 6·23-s − 25-s − 4·29-s − 4·31-s + 2·35-s + 6·37-s + 6·41-s − 4·43-s − 12·47-s + 49-s + 4·53-s + 12·55-s + 4·59-s − 2·61-s − 4·65-s − 4·67-s − 14·71-s − 10·73-s + 6·77-s − 16·79-s + 8·83-s + 10·89-s − 2·91-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.377·7-s − 1.80·11-s + 0.554·13-s + 1.25·23-s − 1/5·25-s − 0.742·29-s − 0.718·31-s + 0.338·35-s + 0.986·37-s + 0.937·41-s − 0.609·43-s − 1.75·47-s + 1/7·49-s + 0.549·53-s + 1.61·55-s + 0.520·59-s − 0.256·61-s − 0.496·65-s − 0.488·67-s − 1.66·71-s − 1.17·73-s + 0.683·77-s − 1.80·79-s + 0.878·83-s + 1.05·89-s − 0.209·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 291312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 291312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 14 T + p T^{2} \) | 1.71.o |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−13.07534779449397, −12.60778431572502, −12.00010484068259, −11.41706255950079, −11.17834232930860, −10.76780029488487, −10.14989680869442, −9.890649375843323, −9.149319672655566, −8.769640435083309, −8.242413195849124, −7.802494627615056, −7.297692161236350, −7.182760155232184, −6.219023297177851, −5.919595705706786, −5.290044648647332, −4.869996387933049, −4.293514492389812, −3.762206949003639, −3.081030945631549, −2.912777193440991, −2.125370338747074, −1.419710926031367, −0.5546811956981816, 0,
0.5546811956981816, 1.419710926031367, 2.125370338747074, 2.912777193440991, 3.081030945631549, 3.762206949003639, 4.293514492389812, 4.869996387933049, 5.290044648647332, 5.919595705706786, 6.219023297177851, 7.182760155232184, 7.297692161236350, 7.802494627615056, 8.242413195849124, 8.769640435083309, 9.149319672655566, 9.890649375843323, 10.14989680869442, 10.76780029488487, 11.17834232930860, 11.41706255950079, 12.00010484068259, 12.60778431572502, 13.07534779449397
