L(s) = 1 | − 2.34·3-s + 3.93·5-s − 4.81·7-s + 2.51·9-s − 2.50·11-s − 4.25·13-s − 9.22·15-s + 17-s + 7.47·19-s + 11.3·21-s − 2.47·23-s + 10.4·25-s + 1.13·27-s + 6.29·29-s + 5.98·31-s + 5.89·33-s − 18.9·35-s + 3.40·37-s + 9.98·39-s + 6.76·41-s − 3.05·43-s + 9.88·45-s − 13.5·47-s + 16.1·49-s − 2.34·51-s − 11.4·53-s − 9.86·55-s + ⋯ |
L(s) = 1 | − 1.35·3-s + 1.75·5-s − 1.81·7-s + 0.838·9-s − 0.756·11-s − 1.17·13-s − 2.38·15-s + 0.242·17-s + 1.71·19-s + 2.46·21-s − 0.516·23-s + 2.08·25-s + 0.219·27-s + 1.16·29-s + 1.07·31-s + 1.02·33-s − 3.19·35-s + 0.559·37-s + 1.59·39-s + 1.05·41-s − 0.465·43-s + 1.47·45-s − 1.96·47-s + 2.30·49-s − 0.328·51-s − 1.57·53-s − 1.32·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
good | 3 | \( 1 + 2.34T + 3T^{2} \) |
| 5 | \( 1 - 3.93T + 5T^{2} \) |
| 7 | \( 1 + 4.81T + 7T^{2} \) |
| 11 | \( 1 + 2.50T + 11T^{2} \) |
| 13 | \( 1 + 4.25T + 13T^{2} \) |
| 19 | \( 1 - 7.47T + 19T^{2} \) |
| 23 | \( 1 + 2.47T + 23T^{2} \) |
| 29 | \( 1 - 6.29T + 29T^{2} \) |
| 31 | \( 1 - 5.98T + 31T^{2} \) |
| 37 | \( 1 - 3.40T + 37T^{2} \) |
| 41 | \( 1 - 6.76T + 41T^{2} \) |
| 43 | \( 1 + 3.05T + 43T^{2} \) |
| 47 | \( 1 + 13.5T + 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 61 | \( 1 + 4.74T + 61T^{2} \) |
| 67 | \( 1 + 7.93T + 67T^{2} \) |
| 71 | \( 1 - 2.82T + 71T^{2} \) |
| 73 | \( 1 + 5.82T + 73T^{2} \) |
| 79 | \( 1 + 8.11T + 79T^{2} \) |
| 83 | \( 1 - 16.8T + 83T^{2} \) |
| 89 | \( 1 + 12.0T + 89T^{2} \) |
| 97 | \( 1 - 9.46T + 97T^{2} \) |
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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
−7.922071598110300615738091835078, −6.95967740396312226258480851349, −6.34065365400095837644931695673, −5.98642762194462548602703367050, −5.25129341617138888318418778476, −4.73975488458831674373445516272, −3.04297878424062566804503340678, −2.65608344146016730979711990289, −1.18695606210322690576879922610, 0,
1.18695606210322690576879922610, 2.65608344146016730979711990289, 3.04297878424062566804503340678, 4.73975488458831674373445516272, 5.25129341617138888318418778476, 5.98642762194462548602703367050, 6.34065365400095837644931695673, 6.95967740396312226258480851349, 7.922071598110300615738091835078