| L(s) = 1 | + 2-s + 3-s − 4-s + 6-s − 3·8-s + 9-s − 12-s + 13-s − 16-s + 18-s − 6·19-s + 4·23-s − 3·24-s + 26-s + 27-s + 2·29-s − 8·31-s + 5·32-s − 36-s + 4·37-s − 6·38-s + 39-s − 6·41-s + 6·43-s + 4·46-s − 8·47-s − 48-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.408·6-s − 1.06·8-s + 1/3·9-s − 0.288·12-s + 0.277·13-s − 1/4·16-s + 0.235·18-s − 1.37·19-s + 0.834·23-s − 0.612·24-s + 0.196·26-s + 0.192·27-s + 0.371·29-s − 1.43·31-s + 0.883·32-s − 1/6·36-s + 0.657·37-s − 0.973·38-s + 0.160·39-s − 0.937·41-s + 0.914·43-s + 0.589·46-s − 1.16·47-s − 0.144·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 281775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 281775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
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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
−12.96139176778118, −12.76151928686641, −12.23460256003825, −11.74908768811403, −11.04008559442754, −10.81336891339766, −10.22205726047459, −9.554939818131871, −9.286848693569349, −8.822438747527718, −8.373259058674022, −7.969754468217054, −7.370833729598948, −6.773547028932044, −6.236646361943234, −5.960172545257792, −5.161996764839468, −4.746004738454106, −4.425732120737948, −3.643714935301152, −3.449033420990576, −2.837068863541672, −2.165631089787945, −1.622657406639731, −0.7494066384434539, 0,
0.7494066384434539, 1.622657406639731, 2.165631089787945, 2.837068863541672, 3.449033420990576, 3.643714935301152, 4.425732120737948, 4.746004738454106, 5.161996764839468, 5.960172545257792, 6.236646361943234, 6.773547028932044, 7.370833729598948, 7.969754468217054, 8.373259058674022, 8.822438747527718, 9.286848693569349, 9.554939818131871, 10.22205726047459, 10.81336891339766, 11.04008559442754, 11.74908768811403, 12.23460256003825, 12.76151928686641, 12.96139176778118
