| L(s) = 1 | + 7-s − 2·11-s − 2·13-s + 17-s + 4·23-s − 5·25-s − 4·29-s + 8·37-s + 2·41-s + 49-s − 2·53-s + 4·59-s − 12·61-s + 8·67-s + 12·71-s − 14·73-s − 2·77-s − 12·79-s + 4·83-s + 6·89-s − 2·91-s + 6·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 0.603·11-s − 0.554·13-s + 0.242·17-s + 0.834·23-s − 25-s − 0.742·29-s + 1.31·37-s + 0.312·41-s + 1/7·49-s − 0.274·53-s + 0.520·59-s − 1.53·61-s + 0.977·67-s + 1.42·71-s − 1.63·73-s − 0.227·77-s − 1.35·79-s + 0.439·83-s + 0.635·89-s − 0.209·91-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-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 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−16.06205139277946, −15.60286592230226, −14.98860190180624, −14.57408895094107, −13.99787969890180, −13.28172481189443, −12.92062318553099, −12.29783756278362, −11.58232196692000, −11.22404989556447, −10.53768686454867, −9.960909230276093, −9.382349949668535, −8.831521554167372, −7.941051557969274, −7.688522173011912, −7.037203781513413, −6.192514736903221, −5.599656049602494, −4.971530805187514, −4.356002678345015, −3.560133253767285, −2.735108253406855, −2.089397074340587, −1.118058515708866, 0,
1.118058515708866, 2.089397074340587, 2.735108253406855, 3.560133253767285, 4.356002678345015, 4.971530805187514, 5.599656049602494, 6.192514736903221, 7.037203781513413, 7.688522173011912, 7.941051557969274, 8.831521554167372, 9.382349949668535, 9.960909230276093, 10.53768686454867, 11.22404989556447, 11.58232196692000, 12.29783756278362, 12.92062318553099, 13.28172481189443, 13.99787969890180, 14.57408895094107, 14.98860190180624, 15.60286592230226, 16.06205139277946
