| L(s) = 1 | + 3-s + 5-s − 7-s + 9-s + 2·13-s + 15-s + 2·17-s − 4·19-s − 21-s + 8·23-s + 25-s + 27-s − 2·29-s − 4·31-s − 35-s + 2·37-s + 2·39-s − 10·41-s + 12·43-s + 45-s − 8·47-s + 49-s + 2·51-s + 53-s − 4·57-s + 8·59-s − 10·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.554·13-s + 0.258·15-s + 0.485·17-s − 0.917·19-s − 0.218·21-s + 1.66·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s − 0.718·31-s − 0.169·35-s + 0.328·37-s + 0.320·39-s − 1.56·41-s + 1.82·43-s + 0.149·45-s − 1.16·47-s + 1/7·49-s + 0.280·51-s + 0.137·53-s − 0.529·57-s + 1.04·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 356160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 356160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 7 | \( 1 + T \) | |
| 53 | \( 1 - T \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 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.ac |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−12.79218545598649, −12.54494052314783, −11.89504359825421, −11.23533815077515, −10.99074567250372, −10.42626543509528, −10.04872568884853, −9.490828396184411, −9.095221639761337, −8.753734815175776, −8.263475175202835, −7.771320878462992, −7.068202494754695, −6.917324799556412, −6.279316065968646, −5.747203334524197, −5.368946197773188, −4.657442831332731, −4.271552340591439, −3.486160761915697, −3.273030015410513, −2.624118000721372, −2.067138149283101, −1.454872939142063, −0.9007795878091718, 0,
0.9007795878091718, 1.454872939142063, 2.067138149283101, 2.624118000721372, 3.273030015410513, 3.486160761915697, 4.271552340591439, 4.657442831332731, 5.368946197773188, 5.747203334524197, 6.279316065968646, 6.917324799556412, 7.068202494754695, 7.771320878462992, 8.263475175202835, 8.753734815175776, 9.095221639761337, 9.490828396184411, 10.04872568884853, 10.42626543509528, 10.99074567250372, 11.23533815077515, 11.89504359825421, 12.54494052314783, 12.79218545598649
