| L(s) = 1 | + 3-s − 7-s + 9-s − 5·13-s − 19-s − 21-s + 6·23-s + 27-s − 5·31-s − 37-s − 5·39-s − 5·43-s + 12·47-s − 6·49-s − 12·53-s − 57-s + 12·59-s + 61-s − 63-s + 13·67-s + 6·69-s − 6·71-s + 2·73-s − 8·79-s + 81-s + 6·83-s − 6·89-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s − 1.38·13-s − 0.229·19-s − 0.218·21-s + 1.25·23-s + 0.192·27-s − 0.898·31-s − 0.164·37-s − 0.800·39-s − 0.762·43-s + 1.75·47-s − 6/7·49-s − 1.64·53-s − 0.132·57-s + 1.56·59-s + 0.128·61-s − 0.125·63-s + 1.58·67-s + 0.722·69-s − 0.712·71-s + 0.234·73-s − 0.900·79-s + 1/9·81-s + 0.658·83-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 17 T + p T^{2} \) | 1.97.ar |
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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
−14.36906954901881, −13.69915141446700, −12.95212873565883, −12.84384550079327, −12.37072566325213, −11.62457521267039, −11.28238393651924, −10.50958046283333, −10.18661179723412, −9.528599014789324, −9.238654374578653, −8.707533905153022, −8.065348562670666, −7.587747570936526, −6.932542953422012, −6.803764124360709, −5.906791523395027, −5.256502611662417, −4.836559464197548, −4.188722975195891, −3.495690667321041, −2.994135513950998, −2.373620636986006, −1.820036883299832, −0.8765382020627382, 0,
0.8765382020627382, 1.820036883299832, 2.373620636986006, 2.994135513950998, 3.495690667321041, 4.188722975195891, 4.836559464197548, 5.256502611662417, 5.906791523395027, 6.803764124360709, 6.932542953422012, 7.587747570936526, 8.065348562670666, 8.707533905153022, 9.238654374578653, 9.528599014789324, 10.18661179723412, 10.50958046283333, 11.28238393651924, 11.62457521267039, 12.37072566325213, 12.84384550079327, 12.95212873565883, 13.69915141446700, 14.36906954901881
