| L(s) = 1 | − 3·9-s − 6·13-s + 2·17-s + 10·29-s − 2·37-s − 10·41-s + 14·53-s − 10·61-s − 6·73-s + 9·81-s − 10·89-s + 18·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 18·117-s + ⋯ |
| L(s) = 1 | − 9-s − 1.66·13-s + 0.485·17-s + 1.85·29-s − 0.328·37-s − 1.56·41-s + 1.92·53-s − 1.28·61-s − 0.702·73-s + 81-s − 1.05·89-s + 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 1.66·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−14.31792285079838, −13.77434618995923, −13.44801462587749, −12.60337825907971, −12.17103418829636, −11.88671678191318, −11.45084241077985, −10.64368987346957, −10.19492724007018, −9.907814121409498, −9.169999806506575, −8.656676478161015, −8.257242426909474, −7.605805759249434, −7.128989352157795, −6.574449485325864, −5.961707883645468, −5.330645064157910, −4.928745942084048, −4.386828509371072, −3.523726426720636, −2.908066433385526, −2.517977903967630, −1.746571399288114, −0.7707478844750073, 0,
0.7707478844750073, 1.746571399288114, 2.517977903967630, 2.908066433385526, 3.523726426720636, 4.386828509371072, 4.928745942084048, 5.330645064157910, 5.961707883645468, 6.574449485325864, 7.128989352157795, 7.605805759249434, 8.257242426909474, 8.656676478161015, 9.169999806506575, 9.907814121409498, 10.19492724007018, 10.64368987346957, 11.45084241077985, 11.88671678191318, 12.17103418829636, 12.60337825907971, 13.44801462587749, 13.77434618995923, 14.31792285079838
