| L(s) = 1 | + 3-s − 2·5-s + 9-s + 6·13-s − 2·15-s − 2·17-s + 4·19-s − 4·23-s − 25-s + 27-s − 2·29-s − 4·31-s − 2·37-s + 6·39-s − 10·41-s + 4·43-s − 2·45-s − 4·47-s − 7·49-s − 2·51-s + 6·53-s + 4·57-s − 12·59-s + 6·61-s − 12·65-s − 4·67-s − 4·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s + 1.66·13-s − 0.516·15-s − 0.485·17-s + 0.917·19-s − 0.834·23-s − 1/5·25-s + 0.192·27-s − 0.371·29-s − 0.718·31-s − 0.328·37-s + 0.960·39-s − 1.56·41-s + 0.609·43-s − 0.298·45-s − 0.583·47-s − 49-s − 0.280·51-s + 0.824·53-s + 0.529·57-s − 1.56·59-s + 0.768·61-s − 1.48·65-s − 0.488·67-s − 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−16.37697290723129, −16.06101218351296, −15.62195865909658, −15.10904395543114, −14.45354569252311, −13.77720551592287, −13.40207923778198, −12.83827649688164, −11.94593334699646, −11.63608247400279, −10.97132811597587, −10.39576110129093, −9.640784193320601, −8.935147714743594, −8.508093780095230, −7.838196402736734, −7.421724762748686, −6.559720589979173, −5.965177947057952, −5.097976104325146, −4.256324146140304, −3.582601156474390, −3.290538978618851, −2.061292076976783, −1.274450454132136, 0,
1.274450454132136, 2.061292076976783, 3.290538978618851, 3.582601156474390, 4.256324146140304, 5.097976104325146, 5.965177947057952, 6.559720589979173, 7.421724762748686, 7.838196402736734, 8.508093780095230, 8.935147714743594, 9.640784193320601, 10.39576110129093, 10.97132811597587, 11.63608247400279, 11.94593334699646, 12.83827649688164, 13.40207923778198, 13.77720551592287, 14.45354569252311, 15.10904395543114, 15.62195865909658, 16.06101218351296, 16.37697290723129
