| L(s) = 1 | − 2·4-s − 5·11-s + 4·16-s − 5·17-s + 19-s + 5·23-s − 5·29-s + 31-s − 10·37-s + 10·43-s + 10·44-s + 5·53-s − 10·59-s − 12·61-s − 8·64-s + 5·67-s + 10·68-s − 2·76-s − 4·79-s + 15·83-s + 15·89-s − 10·92-s − 5·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 4-s − 1.50·11-s + 16-s − 1.21·17-s + 0.229·19-s + 1.04·23-s − 0.928·29-s + 0.179·31-s − 1.64·37-s + 1.52·43-s + 1.50·44-s + 0.686·53-s − 1.30·59-s − 1.53·61-s − 64-s + 0.610·67-s + 1.21·68-s − 0.229·76-s − 0.450·79-s + 1.64·83-s + 1.58·89-s − 1.04·92-s − 0.507·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 341775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 341775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 31 | \( 1 - T \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 5 T + p T^{2} \) | 1.53.af |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 15 T + p T^{2} \) | 1.83.ap |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 + 5 T + p T^{2} \) | 1.97.f |
| 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
−12.90638153715765, −12.39698364794304, −12.10743216068062, −11.25387016019899, −10.92459800983063, −10.52065556705059, −10.16479563663193, −9.489928797173407, −9.049543513599485, −8.838980871498839, −8.279824525627844, −7.623140439260329, −7.491036171985073, −6.845695239398553, −6.144640474403363, −5.709490973834855, −5.152394218672493, −4.805191386693575, −4.439222568059088, −3.646865104173207, −3.310389809380498, −2.597271505043238, −2.129667024827030, −1.347573489453050, −0.5541540387773067, 0,
0.5541540387773067, 1.347573489453050, 2.129667024827030, 2.597271505043238, 3.310389809380498, 3.646865104173207, 4.439222568059088, 4.805191386693575, 5.152394218672493, 5.709490973834855, 6.144640474403363, 6.845695239398553, 7.491036171985073, 7.623140439260329, 8.279824525627844, 8.838980871498839, 9.049543513599485, 9.489928797173407, 10.16479563663193, 10.52065556705059, 10.92459800983063, 11.25387016019899, 12.10743216068062, 12.39698364794304, 12.90638153715765
