| L(s) = 1 | + 1.52·5-s − 2.35·7-s + 0.753·11-s − 4.94·13-s + 0.0567·17-s + 19-s + 8.82·23-s − 2.67·25-s + 4.17·29-s − 0.293·31-s − 3.58·35-s − 8.24·37-s − 4.12·41-s + 5.66·43-s − 0.862·47-s − 1.47·49-s − 4.54·53-s + 1.14·55-s + 6.51·59-s − 2.52·61-s − 7.53·65-s + 1.32·67-s + 4.26·71-s − 0.540·73-s − 1.77·77-s + 13.8·79-s − 16.7·83-s + ⋯ |
| L(s) = 1 | + 0.681·5-s − 0.888·7-s + 0.227·11-s − 1.37·13-s + 0.0137·17-s + 0.229·19-s + 1.83·23-s − 0.535·25-s + 0.775·29-s − 0.0527·31-s − 0.605·35-s − 1.35·37-s − 0.643·41-s + 0.863·43-s − 0.125·47-s − 0.210·49-s − 0.623·53-s + 0.154·55-s + 0.848·59-s − 0.322·61-s − 0.934·65-s + 0.161·67-s + 0.506·71-s − 0.0632·73-s − 0.201·77-s + 1.56·79-s − 1.83·83-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 | \( 1 - 1.52T + 5T^{2} \) |
| 7 | \( 1 + 2.35T + 7T^{2} \) |
| 11 | \( 1 - 0.753T + 11T^{2} \) |
| 13 | \( 1 + 4.94T + 13T^{2} \) |
| 17 | \( 1 - 0.0567T + 17T^{2} \) |
| 23 | \( 1 - 8.82T + 23T^{2} \) |
| 29 | \( 1 - 4.17T + 29T^{2} \) |
| 31 | \( 1 + 0.293T + 31T^{2} \) |
| 37 | \( 1 + 8.24T + 37T^{2} \) |
| 41 | \( 1 + 4.12T + 41T^{2} \) |
| 43 | \( 1 - 5.66T + 43T^{2} \) |
| 47 | \( 1 + 0.862T + 47T^{2} \) |
| 53 | \( 1 + 4.54T + 53T^{2} \) |
| 59 | \( 1 - 6.51T + 59T^{2} \) |
| 61 | \( 1 + 2.52T + 61T^{2} \) |
| 67 | \( 1 - 1.32T + 67T^{2} \) |
| 71 | \( 1 - 4.26T + 71T^{2} \) |
| 73 | \( 1 + 0.540T + 73T^{2} \) |
| 79 | \( 1 - 13.8T + 79T^{2} \) |
| 83 | \( 1 + 16.7T + 83T^{2} \) |
| 89 | \( 1 - 4.82T + 89T^{2} \) |
| 97 | \( 1 + 1.43T + 97T^{2} \) |
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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
−7.24399538458196040626459844642, −6.82876892947601436448373673548, −6.18015135101285267899287654957, −5.26796931571842969878631660280, −4.89193972299045365953481586977, −3.78729002189925693574393372111, −2.98144896633276194063008671576, −2.35836168838504502395018078050, −1.26794760409473292838389370261, 0,
1.26794760409473292838389370261, 2.35836168838504502395018078050, 2.98144896633276194063008671576, 3.78729002189925693574393372111, 4.89193972299045365953481586977, 5.26796931571842969878631660280, 6.18015135101285267899287654957, 6.82876892947601436448373673548, 7.24399538458196040626459844642
