| L(s) = 1 | + 1.35·2-s + 3-s − 0.175·4-s − 0.638·5-s + 1.35·6-s − 0.931·7-s − 2.93·8-s + 9-s − 0.862·10-s − 11-s − 0.175·12-s − 3.34·13-s − 1.25·14-s − 0.638·15-s − 3.61·16-s + 6.34·17-s + 1.35·18-s − 6.01·19-s + 0.111·20-s − 0.931·21-s − 1.35·22-s + 8.35·23-s − 2.93·24-s − 4.59·25-s − 4.52·26-s + 27-s + 0.163·28-s + ⋯ |
| L(s) = 1 | + 0.955·2-s + 0.577·3-s − 0.0875·4-s − 0.285·5-s + 0.551·6-s − 0.351·7-s − 1.03·8-s + 0.333·9-s − 0.272·10-s − 0.301·11-s − 0.0505·12-s − 0.928·13-s − 0.336·14-s − 0.164·15-s − 0.904·16-s + 1.53·17-s + 0.318·18-s − 1.37·19-s + 0.0249·20-s − 0.203·21-s − 0.288·22-s + 1.74·23-s − 0.599·24-s − 0.918·25-s − 0.886·26-s + 0.192·27-s + 0.0308·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 - 1.35T + 2T^{2} \) |
| 5 | \( 1 + 0.638T + 5T^{2} \) |
| 7 | \( 1 + 0.931T + 7T^{2} \) |
| 13 | \( 1 + 3.34T + 13T^{2} \) |
| 17 | \( 1 - 6.34T + 17T^{2} \) |
| 19 | \( 1 + 6.01T + 19T^{2} \) |
| 23 | \( 1 - 8.35T + 23T^{2} \) |
| 29 | \( 1 + 7.15T + 29T^{2} \) |
| 31 | \( 1 + 1.16T + 31T^{2} \) |
| 37 | \( 1 + 7.31T + 37T^{2} \) |
| 41 | \( 1 + 4.22T + 41T^{2} \) |
| 43 | \( 1 + 10.6T + 43T^{2} \) |
| 47 | \( 1 + 13.3T + 47T^{2} \) |
| 53 | \( 1 - 1.73T + 53T^{2} \) |
| 59 | \( 1 + 3.21T + 59T^{2} \) |
| 67 | \( 1 + 0.674T + 67T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 - 10.5T + 73T^{2} \) |
| 79 | \( 1 - 16.8T + 79T^{2} \) |
| 83 | \( 1 - 12.3T + 83T^{2} \) |
| 89 | \( 1 + 3.25T + 89T^{2} \) |
| 97 | \( 1 + 11.7T + 97T^{2} \) |
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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
−8.726655257434644822344120881962, −7.990883670950611586615822141528, −7.14202700846981861278178434476, −6.30877681234903471846973770262, −5.22961860302399943860671896178, −4.78776344071301580916419988868, −3.52295665155159991825144153257, −3.27364904764536361854047805564, −1.97509687389429419950314796208, 0,
1.97509687389429419950314796208, 3.27364904764536361854047805564, 3.52295665155159991825144153257, 4.78776344071301580916419988868, 5.22961860302399943860671896178, 6.30877681234903471846973770262, 7.14202700846981861278178434476, 7.990883670950611586615822141528, 8.726655257434644822344120881962
