| L(s) = 1 | + 2.61·5-s − 4.23·7-s − 3.61·11-s + 13-s − 3.85·17-s + 7.32·19-s + 2.85·23-s + 1.85·25-s + 29-s − 2·31-s − 11.0·35-s − 37-s + 6.23·41-s − 10.0·43-s − 9.85·47-s + 10.9·49-s − 1.38·53-s − 9.47·55-s − 7.94·59-s − 10.2·61-s + 2.61·65-s − 10.3·67-s + 3.70·71-s − 11.5·73-s + 15.3·77-s − 14.8·79-s − 17.1·83-s + ⋯ |
| L(s) = 1 | + 1.17·5-s − 1.60·7-s − 1.09·11-s + 0.277·13-s − 0.934·17-s + 1.68·19-s + 0.595·23-s + 0.370·25-s + 0.185·29-s − 0.359·31-s − 1.87·35-s − 0.164·37-s + 0.973·41-s − 1.53·43-s − 1.43·47-s + 1.56·49-s − 0.189·53-s − 1.27·55-s − 1.03·59-s − 1.31·61-s + 0.324·65-s − 1.26·67-s + 0.440·71-s − 1.35·73-s + 1.74·77-s − 1.67·79-s − 1.88·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 2.61T + 5T^{2} \) |
| 7 | \( 1 + 4.23T + 7T^{2} \) |
| 11 | \( 1 + 3.61T + 11T^{2} \) |
| 17 | \( 1 + 3.85T + 17T^{2} \) |
| 19 | \( 1 - 7.32T + 19T^{2} \) |
| 23 | \( 1 - 2.85T + 23T^{2} \) |
| 29 | \( 1 - T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 - 6.23T + 41T^{2} \) |
| 43 | \( 1 + 10.0T + 43T^{2} \) |
| 47 | \( 1 + 9.85T + 47T^{2} \) |
| 53 | \( 1 + 1.38T + 53T^{2} \) |
| 59 | \( 1 + 7.94T + 59T^{2} \) |
| 61 | \( 1 + 10.2T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 - 3.70T + 71T^{2} \) |
| 73 | \( 1 + 11.5T + 73T^{2} \) |
| 79 | \( 1 + 14.8T + 79T^{2} \) |
| 83 | \( 1 + 17.1T + 83T^{2} \) |
| 89 | \( 1 - 7.70T + 89T^{2} \) |
| 97 | \( 1 + 1.52T + 97T^{2} \) |
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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
−8.609241416722120224802679466631, −7.50772884323708461374972509293, −6.81745747453345634149257496758, −6.05238719207486493032957331064, −5.53377064616618656807797377685, −4.61106039986949671618856297549, −3.19897331917399200934420931823, −2.84489252170349020646264198688, −1.58756337157089266736641612824, 0,
1.58756337157089266736641612824, 2.84489252170349020646264198688, 3.19897331917399200934420931823, 4.61106039986949671618856297549, 5.53377064616618656807797377685, 6.05238719207486493032957331064, 6.81745747453345634149257496758, 7.50772884323708461374972509293, 8.609241416722120224802679466631
