| L(s) = 1 | − 3-s + 5-s − 7-s + 9-s + 6·13-s − 15-s − 3·19-s + 21-s − 6·23-s − 4·25-s − 27-s + 31-s − 35-s + 8·37-s − 6·39-s − 11·41-s − 8·43-s + 45-s − 6·49-s − 4·53-s + 3·57-s − 11·59-s + 8·61-s − 63-s + 6·65-s + 12·67-s + 6·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 1.66·13-s − 0.258·15-s − 0.688·19-s + 0.218·21-s − 1.25·23-s − 4/5·25-s − 0.192·27-s + 0.179·31-s − 0.169·35-s + 1.31·37-s − 0.960·39-s − 1.71·41-s − 1.21·43-s + 0.149·45-s − 6/7·49-s − 0.549·53-s + 0.397·57-s − 1.43·59-s + 1.02·61-s − 0.125·63-s + 0.744·65-s + 1.46·67-s + 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 31 | \( 1 - T \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 11 T + p T^{2} \) | 1.41.l |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + 11 T + p T^{2} \) | 1.59.l |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 7 T + p T^{2} \) | 1.71.ah |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
| 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
−7.912445810306082391381275911414, −6.62558491986575305104413585763, −6.36425929323462385957129672924, −5.76849706640119776763208846067, −4.92165216976348085268554060994, −3.98887766122029595741548257372, −3.41021708212122165614302697250, −2.15291164001475278150774813210, −1.33745920323743149272320932590, 0,
1.33745920323743149272320932590, 2.15291164001475278150774813210, 3.41021708212122165614302697250, 3.98887766122029595741548257372, 4.92165216976348085268554060994, 5.76849706640119776763208846067, 6.36425929323462385957129672924, 6.62558491986575305104413585763, 7.912445810306082391381275911414
