| L(s) = 1 | + 1.53·2-s − 1.68·3-s + 0.370·4-s − 5-s − 2.60·6-s + 7-s − 2.50·8-s − 0.144·9-s − 1.53·10-s + 3.30·11-s − 0.626·12-s − 5.47·13-s + 1.53·14-s + 1.68·15-s − 4.60·16-s − 2.82·17-s − 0.222·18-s + 7.71·19-s − 0.370·20-s − 1.68·21-s + 5.08·22-s + 0.0524·23-s + 4.23·24-s + 25-s − 8.43·26-s + 5.31·27-s + 0.370·28-s + ⋯ |
| L(s) = 1 | + 1.08·2-s − 0.975·3-s + 0.185·4-s − 0.447·5-s − 1.06·6-s + 0.377·7-s − 0.886·8-s − 0.0482·9-s − 0.486·10-s + 0.995·11-s − 0.180·12-s − 1.51·13-s + 0.411·14-s + 0.436·15-s − 1.15·16-s − 0.685·17-s − 0.0524·18-s + 1.76·19-s − 0.0829·20-s − 0.368·21-s + 1.08·22-s + 0.0109·23-s + 0.865·24-s + 0.200·25-s − 1.65·26-s + 1.02·27-s + 0.0700·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 229 | \( 1 - T \) |
| good | 2 | \( 1 - 1.53T + 2T^{2} \) |
| 3 | \( 1 + 1.68T + 3T^{2} \) |
| 11 | \( 1 - 3.30T + 11T^{2} \) |
| 13 | \( 1 + 5.47T + 13T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 - 7.71T + 19T^{2} \) |
| 23 | \( 1 - 0.0524T + 23T^{2} \) |
| 29 | \( 1 - 1.13T + 29T^{2} \) |
| 31 | \( 1 - 3.82T + 31T^{2} \) |
| 37 | \( 1 - 2.49T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 - 1.69T + 43T^{2} \) |
| 47 | \( 1 + 1.99T + 47T^{2} \) |
| 53 | \( 1 + 5.10T + 53T^{2} \) |
| 59 | \( 1 + 3.60T + 59T^{2} \) |
| 61 | \( 1 + 3.03T + 61T^{2} \) |
| 67 | \( 1 + 11.8T + 67T^{2} \) |
| 71 | \( 1 + 11.7T + 71T^{2} \) |
| 73 | \( 1 - 3.56T + 73T^{2} \) |
| 79 | \( 1 + 5.02T + 79T^{2} \) |
| 83 | \( 1 - 0.391T + 83T^{2} \) |
| 89 | \( 1 - 10.5T + 89T^{2} \) |
| 97 | \( 1 + 6.09T + 97T^{2} \) |
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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
−7.28600252610429629607777373919, −6.56249054963314600462878892249, −5.92308216789517907892796579819, −5.28428840497192599018243913921, −4.62718258309010576034932490504, −4.31064749182113010913500052401, −3.20852954280098867141655579254, −2.57765193186345553727213628193, −1.10294371816534799102149040518, 0,
1.10294371816534799102149040518, 2.57765193186345553727213628193, 3.20852954280098867141655579254, 4.31064749182113010913500052401, 4.62718258309010576034932490504, 5.28428840497192599018243913921, 5.92308216789517907892796579819, 6.56249054963314600462878892249, 7.28600252610429629607777373919
