| L(s) = 1 | + 5-s − 3·7-s − 5·11-s − 5·13-s + 2·17-s − 4·19-s + 23-s − 4·25-s + 9·29-s − 31-s − 3·35-s − 6·37-s − 3·41-s + 43-s + 3·47-s + 2·49-s − 2·53-s − 5·55-s − 11·59-s + 7·61-s − 5·65-s − 67-s − 4·71-s − 2·73-s + 15·77-s + 79-s − 83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.13·7-s − 1.50·11-s − 1.38·13-s + 0.485·17-s − 0.917·19-s + 0.208·23-s − 4/5·25-s + 1.67·29-s − 0.179·31-s − 0.507·35-s − 0.986·37-s − 0.468·41-s + 0.152·43-s + 0.437·47-s + 2/7·49-s − 0.274·53-s − 0.674·55-s − 1.43·59-s + 0.896·61-s − 0.620·65-s − 0.122·67-s − 0.474·71-s − 0.234·73-s + 1.70·77-s + 0.112·79-s − 0.109·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 11 T + p T^{2} \) | 1.59.l |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 + T + p T^{2} \) | 1.67.b |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 + T + p T^{2} \) | 1.83.b |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 + 13 T + p T^{2} \) | 1.97.n |
| 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
−10.13681284927785547962865876897, −9.489148932003317588238101757941, −8.359584896208198912892556811654, −7.43764095199028116947875415823, −6.53308020646296798141338084684, −5.57057938485456350288330386239, −4.67089571149572557248085150675, −3.15917888226170604209305923594, −2.30173437261677938790892627190, 0,
2.30173437261677938790892627190, 3.15917888226170604209305923594, 4.67089571149572557248085150675, 5.57057938485456350288330386239, 6.53308020646296798141338084684, 7.43764095199028116947875415823, 8.359584896208198912892556811654, 9.489148932003317588238101757941, 10.13681284927785547962865876897
