| L(s) = 1 | − 5-s + 4·7-s + 2·13-s + 4·19-s + 8·23-s + 25-s − 6·29-s − 4·35-s + 6·37-s + 2·41-s + 4·43-s − 8·47-s + 9·49-s − 10·53-s − 8·59-s − 6·61-s − 2·65-s + 4·67-s − 8·71-s − 10·73-s − 12·83-s + 14·89-s + 8·91-s − 4·95-s − 10·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.51·7-s + 0.554·13-s + 0.917·19-s + 1.66·23-s + 1/5·25-s − 1.11·29-s − 0.676·35-s + 0.986·37-s + 0.312·41-s + 0.609·43-s − 1.16·47-s + 9/7·49-s − 1.37·53-s − 1.04·59-s − 0.768·61-s − 0.248·65-s + 0.488·67-s − 0.949·71-s − 1.17·73-s − 1.31·83-s + 1.48·89-s + 0.838·91-s − 0.410·95-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−14.00935688446935, −13.54243342069118, −12.81413222247884, −12.69441271163800, −11.73893096288824, −11.48353807666776, −11.09599004020996, −10.80412355629218, −10.06383339751557, −9.375686375751550, −8.992133360118018, −8.487149803829824, −7.876873898615414, −7.518673592078692, −7.186655885356755, −6.297338531307339, −5.829538992895116, −5.114291922676678, −4.765958737770459, −4.285509080327362, −3.493517157053114, −3.030884109027898, −2.260498050200141, −1.281233251200247, −1.242678935596306, 0,
1.242678935596306, 1.281233251200247, 2.260498050200141, 3.030884109027898, 3.493517157053114, 4.285509080327362, 4.765958737770459, 5.114291922676678, 5.829538992895116, 6.297338531307339, 7.186655885356755, 7.518673592078692, 7.876873898615414, 8.487149803829824, 8.992133360118018, 9.375686375751550, 10.06383339751557, 10.80412355629218, 11.09599004020996, 11.48353807666776, 11.73893096288824, 12.69441271163800, 12.81413222247884, 13.54243342069118, 14.00935688446935
