| L(s) = 1 | + 3-s − 1.23·5-s + 3.23·7-s + 9-s − 4·11-s + 4.47·13-s − 1.23·15-s − 2.76·17-s − 7.23·19-s + 3.23·21-s + 23-s − 3.47·25-s + 27-s − 4.47·29-s − 6.47·31-s − 4·33-s − 4.00·35-s − 4.47·37-s + 4.47·39-s − 10.9·41-s + 5.70·43-s − 1.23·45-s − 4·47-s + 3.47·49-s − 2.76·51-s + 5.23·53-s + 4.94·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.552·5-s + 1.22·7-s + 0.333·9-s − 1.20·11-s + 1.24·13-s − 0.319·15-s − 0.670·17-s − 1.66·19-s + 0.706·21-s + 0.208·23-s − 0.694·25-s + 0.192·27-s − 0.830·29-s − 1.16·31-s − 0.696·33-s − 0.676·35-s − 0.735·37-s + 0.716·39-s − 1.70·41-s + 0.870·43-s − 0.184·45-s − 0.583·47-s + 0.496·49-s − 0.387·51-s + 0.719·53-s + 0.666·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 1.23T + 5T^{2} \) |
| 7 | \( 1 - 3.23T + 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 + 2.76T + 17T^{2} \) |
| 19 | \( 1 + 7.23T + 19T^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 + 6.47T + 31T^{2} \) |
| 37 | \( 1 + 4.47T + 37T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 - 5.70T + 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 - 5.23T + 53T^{2} \) |
| 59 | \( 1 - 4.94T + 59T^{2} \) |
| 61 | \( 1 + 4.47T + 61T^{2} \) |
| 67 | \( 1 + 0.763T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 6.94T + 73T^{2} \) |
| 79 | \( 1 - 9.70T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + 1.23T + 89T^{2} \) |
| 97 | \( 1 - 8.47T + 97T^{2} \) |
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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
−8.115997781253412285452799498290, −7.50248475251985106727903395171, −6.67169948047704505004965306015, −5.68658064749301217449407748682, −4.92608224608797885293664641535, −4.12259063501064428436824062782, −3.50872506551610863095515269526, −2.27377589777439877721660510282, −1.66213090439218947759706895598, 0,
1.66213090439218947759706895598, 2.27377589777439877721660510282, 3.50872506551610863095515269526, 4.12259063501064428436824062782, 4.92608224608797885293664641535, 5.68658064749301217449407748682, 6.67169948047704505004965306015, 7.50248475251985106727903395171, 8.115997781253412285452799498290
