| L(s) = 1 | − 2·2-s + 2·4-s + 11-s + 3·13-s − 4·16-s − 6·19-s − 2·22-s − 3·23-s − 6·26-s + 3·29-s + 8·32-s − 2·37-s + 12·38-s + 3·41-s + 43-s + 2·44-s + 6·46-s − 47-s + 6·52-s + 53-s − 6·58-s − 6·59-s − 4·61-s − 8·64-s − 8·67-s + 8·71-s − 10·73-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 0.301·11-s + 0.832·13-s − 16-s − 1.37·19-s − 0.426·22-s − 0.625·23-s − 1.17·26-s + 0.557·29-s + 1.41·32-s − 0.328·37-s + 1.94·38-s + 0.468·41-s + 0.152·43-s + 0.301·44-s + 0.884·46-s − 0.145·47-s + 0.832·52-s + 0.137·53-s − 0.787·58-s − 0.781·59-s − 0.512·61-s − 64-s − 0.977·67-s + 0.949·71-s − 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 2 | \( 1 + p T + p T^{2} \) | 1.2.c |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
| 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
−13.88568923890727, −13.21407849262685, −12.89418650352833, −12.17604082430049, −11.73497233047112, −11.18972924718218, −10.71912871587299, −10.31988520217184, −10.00247979487886, −9.146953123143982, −8.972382110593871, −8.552842765649689, −7.847377670998678, −7.719794156069373, −6.873903972367098, −6.411150571515992, −6.091922853548752, −5.289441447133890, −4.470267142505765, −4.185971995804013, −3.421027376329679, −2.659896184657409, −1.966845695038652, −1.492933258005265, −0.7450747460918425, 0,
0.7450747460918425, 1.492933258005265, 1.966845695038652, 2.659896184657409, 3.421027376329679, 4.185971995804013, 4.470267142505765, 5.289441447133890, 6.091922853548752, 6.411150571515992, 6.873903972367098, 7.719794156069373, 7.847377670998678, 8.552842765649689, 8.972382110593871, 9.146953123143982, 10.00247979487886, 10.31988520217184, 10.71912871587299, 11.18972924718218, 11.73497233047112, 12.17604082430049, 12.89418650352833, 13.21407849262685, 13.88568923890727
