| L(s) = 1 | + 3-s − 4·7-s + 9-s + 13-s + 4·19-s − 4·21-s − 5·25-s + 27-s + 10·31-s − 2·37-s + 39-s + 6·41-s + 10·43-s + 9·49-s − 6·53-s + 4·57-s + 2·61-s − 4·63-s + 2·67-s − 12·71-s + 10·73-s − 5·75-s − 10·79-s + 81-s + 12·83-s + 12·89-s − 4·91-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.51·7-s + 1/3·9-s + 0.277·13-s + 0.917·19-s − 0.872·21-s − 25-s + 0.192·27-s + 1.79·31-s − 0.328·37-s + 0.160·39-s + 0.937·41-s + 1.52·43-s + 9/7·49-s − 0.824·53-s + 0.529·57-s + 0.256·61-s − 0.503·63-s + 0.244·67-s − 1.42·71-s + 1.17·73-s − 0.577·75-s − 1.12·79-s + 1/9·81-s + 1.31·83-s + 1.27·89-s − 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 302016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 302016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.06406512791017, −12.35142111952872, −12.20672889482915, −11.60887404256848, −11.06258384169154, −10.48516941801663, −10.00999945367598, −9.722882017188817, −9.202599876851798, −8.986632867545131, −8.206582873264360, −7.850402139386665, −7.337693864649036, −6.878012121618918, −6.176556333714210, −6.115129522000276, −5.419842289880881, −4.719691963179884, −4.182677413875832, −3.614996927956396, −3.283840144048288, −2.611592295932710, −2.354924859551903, −1.342030154065384, −0.8169894289225903, 0,
0.8169894289225903, 1.342030154065384, 2.354924859551903, 2.611592295932710, 3.283840144048288, 3.614996927956396, 4.182677413875832, 4.719691963179884, 5.419842289880881, 6.115129522000276, 6.176556333714210, 6.878012121618918, 7.337693864649036, 7.850402139386665, 8.206582873264360, 8.986632867545131, 9.202599876851798, 9.722882017188817, 10.00999945367598, 10.48516941801663, 11.06258384169154, 11.60887404256848, 12.20672889482915, 12.35142111952872, 13.06406512791017
