| L(s) = 1 | + 2-s − 3-s + 4-s − 2·5-s − 6-s + 8-s + 9-s − 2·10-s − 11-s − 12-s − 2·13-s + 2·15-s + 16-s − 4·17-s + 18-s − 4·19-s − 2·20-s − 22-s + 6·23-s − 24-s − 25-s − 2·26-s − 27-s − 4·29-s + 2·30-s + 4·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.632·10-s − 0.301·11-s − 0.288·12-s − 0.554·13-s + 0.516·15-s + 1/4·16-s − 0.970·17-s + 0.235·18-s − 0.917·19-s − 0.447·20-s − 0.213·22-s + 1.25·23-s − 0.204·24-s − 1/5·25-s − 0.392·26-s − 0.192·27-s − 0.742·29-s + 0.365·30-s + 0.718·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110946 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110946 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 41 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 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.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−14.02992154984936, −13.57178470829242, −13.06841268647931, −12.61203581518827, −12.31866046705075, −11.62032083143960, −11.32235512068526, −10.95951698474710, −10.37965751611408, −9.934234691578096, −9.096374730362575, −8.777786837084450, −7.922598235399074, −7.668466550841951, −7.036795019912079, −6.544052624895389, −6.154751892233313, −5.382793778922116, −4.906442785885257, −4.363185639593088, −4.136169468243656, −3.138646975501830, −2.888833529836203, −1.928196205567341, −1.364573931008364, 0, 0,
1.364573931008364, 1.928196205567341, 2.888833529836203, 3.138646975501830, 4.136169468243656, 4.363185639593088, 4.906442785885257, 5.382793778922116, 6.154751892233313, 6.544052624895389, 7.036795019912079, 7.668466550841951, 7.922598235399074, 8.777786837084450, 9.096374730362575, 9.934234691578096, 10.37965751611408, 10.95951698474710, 11.32235512068526, 11.62032083143960, 12.31866046705075, 12.61203581518827, 13.06841268647931, 13.57178470829242, 14.02992154984936
