| L(s) = 1 | − 1.43·3-s − 2.34·5-s − 0.938·9-s + 5.38·11-s + 5.32·13-s + 3.36·15-s − 7.88·17-s − 6.06·19-s − 23-s + 0.486·25-s + 5.65·27-s + 1.87·29-s + 5.83·31-s − 7.73·33-s + 1.19·37-s − 7.64·39-s − 6.45·41-s − 6.93·43-s + 2.19·45-s + 10.8·47-s + 11.3·51-s + 7.94·53-s − 12.6·55-s + 8.70·57-s + 2.69·59-s + 3.27·61-s − 12.4·65-s + ⋯ |
| L(s) = 1 | − 0.828·3-s − 1.04·5-s − 0.312·9-s + 1.62·11-s + 1.47·13-s + 0.868·15-s − 1.91·17-s − 1.39·19-s − 0.208·23-s + 0.0973·25-s + 1.08·27-s + 0.348·29-s + 1.04·31-s − 1.34·33-s + 0.195·37-s − 1.22·39-s − 1.00·41-s − 1.05·43-s + 0.327·45-s + 1.58·47-s + 1.58·51-s + 1.09·53-s − 1.70·55-s + 1.15·57-s + 0.350·59-s + 0.419·61-s − 1.54·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4508 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4508 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 1.43T + 3T^{2} \) |
| 5 | \( 1 + 2.34T + 5T^{2} \) |
| 11 | \( 1 - 5.38T + 11T^{2} \) |
| 13 | \( 1 - 5.32T + 13T^{2} \) |
| 17 | \( 1 + 7.88T + 17T^{2} \) |
| 19 | \( 1 + 6.06T + 19T^{2} \) |
| 29 | \( 1 - 1.87T + 29T^{2} \) |
| 31 | \( 1 - 5.83T + 31T^{2} \) |
| 37 | \( 1 - 1.19T + 37T^{2} \) |
| 41 | \( 1 + 6.45T + 41T^{2} \) |
| 43 | \( 1 + 6.93T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 - 7.94T + 53T^{2} \) |
| 59 | \( 1 - 2.69T + 59T^{2} \) |
| 61 | \( 1 - 3.27T + 61T^{2} \) |
| 67 | \( 1 + 1.82T + 67T^{2} \) |
| 71 | \( 1 - 7.30T + 71T^{2} \) |
| 73 | \( 1 - 7.51T + 73T^{2} \) |
| 79 | \( 1 + 2.49T + 79T^{2} \) |
| 83 | \( 1 - 16.8T + 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 + 11.9T + 97T^{2} \) |
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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
−8.286999860386157778846961845306, −6.88096217031116004108600852429, −6.52962479671568020840846737282, −6.04974061870682262720632115520, −4.88140894198853507059080503597, −4.03624831561775571646091411115, −3.80700011708833684171317244059, −2.38056940886870610253762671777, −1.11751148130416146385011373280, 0,
1.11751148130416146385011373280, 2.38056940886870610253762671777, 3.80700011708833684171317244059, 4.03624831561775571646091411115, 4.88140894198853507059080503597, 6.04974061870682262720632115520, 6.52962479671568020840846737282, 6.88096217031116004108600852429, 8.286999860386157778846961845306
