| L(s) = 1 | + 2·7-s − 4·11-s + 13-s − 2·17-s − 4·19-s + 4·23-s + 6·31-s + 2·37-s + 6·41-s + 12·47-s − 3·49-s + 8·53-s − 12·59-s + 6·61-s + 12·71-s − 6·73-s − 8·77-s − 14·79-s − 4·83-s + 18·89-s + 2·91-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 1.20·11-s + 0.277·13-s − 0.485·17-s − 0.917·19-s + 0.834·23-s + 1.07·31-s + 0.328·37-s + 0.937·41-s + 1.75·47-s − 3/7·49-s + 1.09·53-s − 1.56·59-s + 0.768·61-s + 1.42·71-s − 0.702·73-s − 0.911·77-s − 1.57·79-s − 0.439·83-s + 1.90·89-s + 0.209·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 374400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 374400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.711608325\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.711608325\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 8 T + p T^{2} \) | 1.53.ai |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−12.55782600802229, −12.01074954462399, −11.54547687603813, −10.94483788451254, −10.86424358626033, −10.28830592458485, −9.926896551265018, −9.184254100472052, −8.754523373136730, −8.490356983036422, −7.784846377284849, −7.637417637720451, −7.023297884849317, −6.413257204671098, −6.024080865208640, −5.405671365396262, −5.021122741322506, −4.412109174718858, −4.198506325389412, −3.377031877163829, −2.716142646172273, −2.391328045905099, −1.804104073416149, −1.009540295510378, −0.4733763145762357,
0.4733763145762357, 1.009540295510378, 1.804104073416149, 2.391328045905099, 2.716142646172273, 3.377031877163829, 4.198506325389412, 4.412109174718858, 5.021122741322506, 5.405671365396262, 6.024080865208640, 6.413257204671098, 7.023297884849317, 7.637417637720451, 7.784846377284849, 8.490356983036422, 8.754523373136730, 9.184254100472052, 9.926896551265018, 10.28830592458485, 10.86424358626033, 10.94483788451254, 11.54547687603813, 12.01074954462399, 12.55782600802229
