| L(s) = 1 | + 3.69·5-s + 1.58·7-s − 4.58·11-s + 1.58·13-s − 5.57·17-s + 19-s − 9.09·23-s + 8.68·25-s − 7.65·29-s + 2.58·31-s + 5.87·35-s + 0.287·37-s + 3.69·41-s − 9.32·43-s − 9·47-s − 4.47·49-s + 8.24·53-s − 16.9·55-s − 7.46·59-s + 0.189·61-s + 5.87·65-s − 8.08·67-s + 6.21·71-s − 1.22·73-s − 7.28·77-s + 5.90·79-s + 2.84·83-s + ⋯ |
| L(s) = 1 | + 1.65·5-s + 0.600·7-s − 1.38·11-s + 0.440·13-s − 1.35·17-s + 0.229·19-s − 1.89·23-s + 1.73·25-s − 1.42·29-s + 0.464·31-s + 0.993·35-s + 0.0473·37-s + 0.577·41-s − 1.42·43-s − 1.31·47-s − 0.639·49-s + 1.13·53-s − 2.28·55-s − 0.971·59-s + 0.0242·61-s + 0.728·65-s − 0.987·67-s + 0.737·71-s − 0.143·73-s − 0.830·77-s + 0.664·79-s + 0.312·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 3.69T + 5T^{2} \) |
| 7 | \( 1 - 1.58T + 7T^{2} \) |
| 11 | \( 1 + 4.58T + 11T^{2} \) |
| 13 | \( 1 - 1.58T + 13T^{2} \) |
| 17 | \( 1 + 5.57T + 17T^{2} \) |
| 23 | \( 1 + 9.09T + 23T^{2} \) |
| 29 | \( 1 + 7.65T + 29T^{2} \) |
| 31 | \( 1 - 2.58T + 31T^{2} \) |
| 37 | \( 1 - 0.287T + 37T^{2} \) |
| 41 | \( 1 - 3.69T + 41T^{2} \) |
| 43 | \( 1 + 9.32T + 43T^{2} \) |
| 47 | \( 1 + 9T + 47T^{2} \) |
| 53 | \( 1 - 8.24T + 53T^{2} \) |
| 59 | \( 1 + 7.46T + 59T^{2} \) |
| 61 | \( 1 - 0.189T + 61T^{2} \) |
| 67 | \( 1 + 8.08T + 67T^{2} \) |
| 71 | \( 1 - 6.21T + 71T^{2} \) |
| 73 | \( 1 + 1.22T + 73T^{2} \) |
| 79 | \( 1 - 5.90T + 79T^{2} \) |
| 83 | \( 1 - 2.84T + 83T^{2} \) |
| 89 | \( 1 - 1.03T + 89T^{2} \) |
| 97 | \( 1 + 15.4T + 97T^{2} \) |
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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
−7.57445294951278136289235993204, −6.53704874342248577202271052285, −6.10113400285805114962022242568, −5.36452819839198271784151409429, −4.90745179926347331953889903265, −3.96422196801138386009707615783, −2.81888451510527336796820426260, −2.07911804870742347109711674685, −1.62726411188950480625219923800, 0,
1.62726411188950480625219923800, 2.07911804870742347109711674685, 2.81888451510527336796820426260, 3.96422196801138386009707615783, 4.90745179926347331953889903265, 5.36452819839198271784151409429, 6.10113400285805114962022242568, 6.53704874342248577202271052285, 7.57445294951278136289235993204
