| L(s) = 1 | − 3-s + 9-s + 13-s − 2·17-s − 4·19-s − 27-s − 6·29-s − 2·37-s − 39-s + 6·41-s + 12·43-s − 4·47-s − 7·49-s + 2·51-s + 6·53-s + 4·57-s − 8·59-s + 2·61-s − 4·67-s + 12·71-s + 14·73-s + 81-s − 8·83-s + 6·87-s − 18·89-s + 6·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 0.277·13-s − 0.485·17-s − 0.917·19-s − 0.192·27-s − 1.11·29-s − 0.328·37-s − 0.160·39-s + 0.937·41-s + 1.82·43-s − 0.583·47-s − 49-s + 0.280·51-s + 0.824·53-s + 0.529·57-s − 1.04·59-s + 0.256·61-s − 0.488·67-s + 1.42·71-s + 1.63·73-s + 1/9·81-s − 0.878·83-s + 0.643·87-s − 1.90·89-s + 0.609·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.47701945669915, −14.04593506645357, −13.44643749144378, −12.84718010807630, −12.60579245966283, −12.05646383561304, −11.30259063717428, −10.93540448627591, −10.76747670337138, −9.811766613260765, −9.561918930519805, −8.828262896289255, −8.414731036051254, −7.668130112589252, −7.263452214386962, −6.529542135844435, −6.176440566950810, −5.559416739863593, −5.021181900532788, −4.244153348133357, −3.967342238209474, −3.101646870203020, −2.309213442101535, −1.729730826601354, −0.8270383113808422, 0,
0.8270383113808422, 1.729730826601354, 2.309213442101535, 3.101646870203020, 3.967342238209474, 4.244153348133357, 5.021181900532788, 5.559416739863593, 6.176440566950810, 6.529542135844435, 7.263452214386962, 7.668130112589252, 8.414731036051254, 8.828262896289255, 9.561918930519805, 9.811766613260765, 10.76747670337138, 10.93540448627591, 11.30259063717428, 12.05646383561304, 12.60579245966283, 12.84718010807630, 13.44643749144378, 14.04593506645357, 14.47701945669915
