| L(s) = 1 | + 2·5-s − 2·7-s − 11-s − 13-s + 7·17-s − 6·19-s − 9·23-s − 25-s − 4·29-s + 31-s − 4·35-s − 4·37-s + 11·41-s + 43-s − 3·49-s − 11·53-s − 2·55-s − 12·59-s − 2·65-s + 7·67-s + 10·71-s − 4·73-s + 2·77-s − 8·79-s + 3·83-s + 14·85-s − 6·89-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.755·7-s − 0.301·11-s − 0.277·13-s + 1.69·17-s − 1.37·19-s − 1.87·23-s − 1/5·25-s − 0.742·29-s + 0.179·31-s − 0.676·35-s − 0.657·37-s + 1.71·41-s + 0.152·43-s − 3/7·49-s − 1.51·53-s − 0.269·55-s − 1.56·59-s − 0.248·65-s + 0.855·67-s + 1.18·71-s − 0.468·73-s + 0.227·77-s − 0.900·79-s + 0.329·83-s + 1.51·85-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 43 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 9 T + p T^{2} \) | 1.23.j |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 11 T + p T^{2} \) | 1.41.al |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 11 T + p T^{2} \) | 1.53.l |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 3 T + p T^{2} \) | 1.97.ad |
| 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
−8.172798085040851434259865016352, −7.73051961878322875646964190327, −6.63888697188311106294994263711, −5.96481744627945684012587539218, −5.54430852948314662565357628228, −4.36868696173016823556151509494, −3.49782700153426051498245034066, −2.50835017996577444889153230954, −1.63770359422184508311135901286, 0,
1.63770359422184508311135901286, 2.50835017996577444889153230954, 3.49782700153426051498245034066, 4.36868696173016823556151509494, 5.54430852948314662565357628228, 5.96481744627945684012587539218, 6.63888697188311106294994263711, 7.73051961878322875646964190327, 8.172798085040851434259865016352
