| L(s) = 1 | − 2-s + 4-s + 3.46·5-s − 1.73·7-s − 8-s − 3.46·10-s − 5.19·13-s + 1.73·14-s + 16-s − 3.46·19-s + 3.46·20-s + 6.92·23-s + 6.99·25-s + 5.19·26-s − 1.73·28-s − 6·29-s − 5·31-s − 32-s − 5.99·35-s − 2·37-s + 3.46·38-s − 3.46·40-s + 6·41-s + 10.3·43-s − 6.92·46-s − 10.3·47-s − 4·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.54·5-s − 0.654·7-s − 0.353·8-s − 1.09·10-s − 1.44·13-s + 0.462·14-s + 0.250·16-s − 0.794·19-s + 0.774·20-s + 1.44·23-s + 1.39·25-s + 1.01·26-s − 0.327·28-s − 1.11·29-s − 0.898·31-s − 0.176·32-s − 1.01·35-s − 0.328·37-s + 0.561·38-s − 0.547·40-s + 0.937·41-s + 1.58·43-s − 1.02·46-s − 1.51·47-s − 0.571·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 3.46T + 5T^{2} \) |
| 7 | \( 1 + 1.73T + 7T^{2} \) |
| 13 | \( 1 + 5.19T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 3.46T + 19T^{2} \) |
| 23 | \( 1 - 6.92T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 5T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 3.46T + 59T^{2} \) |
| 61 | \( 1 - 13.8T + 61T^{2} \) |
| 67 | \( 1 - 13T + 67T^{2} \) |
| 71 | \( 1 + 3.46T + 71T^{2} \) |
| 73 | \( 1 - 1.73T + 73T^{2} \) |
| 79 | \( 1 + 8.66T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + 17.3T + 89T^{2} \) |
| 97 | \( 1 + 19T + 97T^{2} \) |
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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
−7.55903094574691272356948295259, −6.90367566318464986429032124418, −6.43945749721904990976223323552, −5.51953852873453253940016016869, −5.14261142488494002687788460891, −3.90969840300634673846210866791, −2.72727524437429482645411959495, −2.32144622473334451970669999271, −1.35275361334198414807186134396, 0,
1.35275361334198414807186134396, 2.32144622473334451970669999271, 2.72727524437429482645411959495, 3.90969840300634673846210866791, 5.14261142488494002687788460891, 5.51953852873453253940016016869, 6.43945749721904990976223323552, 6.90367566318464986429032124418, 7.55903094574691272356948295259
