| L(s) = 1 | − 0.531·3-s − 0.110·5-s + 7-s − 2.71·9-s + 3.70·11-s + 0.433·13-s + 0.0588·15-s − 17-s + 0.814·19-s − 0.531·21-s + 1.18·23-s − 4.98·25-s + 3.04·27-s − 7.33·29-s + 2.31·31-s − 1.97·33-s − 0.110·35-s − 3.22·37-s − 0.230·39-s + 9.96·41-s − 12.2·43-s + 0.300·45-s + 4.33·47-s + 49-s + 0.531·51-s − 0.931·53-s − 0.409·55-s + ⋯ |
| L(s) = 1 | − 0.307·3-s − 0.0494·5-s + 0.377·7-s − 0.905·9-s + 1.11·11-s + 0.120·13-s + 0.0151·15-s − 0.242·17-s + 0.186·19-s − 0.116·21-s + 0.247·23-s − 0.997·25-s + 0.585·27-s − 1.36·29-s + 0.416·31-s − 0.343·33-s − 0.0186·35-s − 0.530·37-s − 0.0368·39-s + 1.55·41-s − 1.87·43-s + 0.0447·45-s + 0.632·47-s + 0.142·49-s + 0.0744·51-s − 0.127·53-s − 0.0552·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 0.531T + 3T^{2} \) |
| 5 | \( 1 + 0.110T + 5T^{2} \) |
| 11 | \( 1 - 3.70T + 11T^{2} \) |
| 13 | \( 1 - 0.433T + 13T^{2} \) |
| 19 | \( 1 - 0.814T + 19T^{2} \) |
| 23 | \( 1 - 1.18T + 23T^{2} \) |
| 29 | \( 1 + 7.33T + 29T^{2} \) |
| 31 | \( 1 - 2.31T + 31T^{2} \) |
| 37 | \( 1 + 3.22T + 37T^{2} \) |
| 41 | \( 1 - 9.96T + 41T^{2} \) |
| 43 | \( 1 + 12.2T + 43T^{2} \) |
| 47 | \( 1 - 4.33T + 47T^{2} \) |
| 53 | \( 1 + 0.931T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 - 5.63T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 + 4.54T + 71T^{2} \) |
| 73 | \( 1 - 8.09T + 73T^{2} \) |
| 79 | \( 1 - 6.73T + 79T^{2} \) |
| 83 | \( 1 + 4.28T + 83T^{2} \) |
| 89 | \( 1 + 8.11T + 89T^{2} \) |
| 97 | \( 1 - 0.418T + 97T^{2} \) |
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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
−7.54913671309560518644291479946, −6.77018161565476296313353732985, −6.06482218236725101006911956819, −5.55323977902771613922696599717, −4.70782659288297399456974268173, −3.93296943149291171641046669652, −3.21294208210372810262506848809, −2.16623588440482689769486763103, −1.26457153626457335169877345929, 0,
1.26457153626457335169877345929, 2.16623588440482689769486763103, 3.21294208210372810262506848809, 3.93296943149291171641046669652, 4.70782659288297399456974268173, 5.55323977902771613922696599717, 6.06482218236725101006911956819, 6.77018161565476296313353732985, 7.54913671309560518644291479946
