| L(s) = 1 | − 0.175·2-s − 3-s − 1.96·4-s − 1.94·5-s + 0.175·6-s − 2.33·7-s + 0.696·8-s + 9-s + 0.341·10-s + 11-s + 1.96·12-s + 1.20·13-s + 0.410·14-s + 1.94·15-s + 3.81·16-s + 2.45·17-s − 0.175·18-s + 3.98·19-s + 3.83·20-s + 2.33·21-s − 0.175·22-s − 3.96·23-s − 0.696·24-s − 1.21·25-s − 0.211·26-s − 27-s + 4.60·28-s + ⋯ |
| L(s) = 1 | − 0.124·2-s − 0.577·3-s − 0.984·4-s − 0.870·5-s + 0.0716·6-s − 0.883·7-s + 0.246·8-s + 0.333·9-s + 0.107·10-s + 0.301·11-s + 0.568·12-s + 0.333·13-s + 0.109·14-s + 0.502·15-s + 0.954·16-s + 0.595·17-s − 0.0413·18-s + 0.915·19-s + 0.857·20-s + 0.510·21-s − 0.0373·22-s − 0.826·23-s − 0.142·24-s − 0.242·25-s − 0.0413·26-s − 0.192·27-s + 0.870·28-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 | \( 1 + 0.175T + 2T^{2} \) |
| 5 | \( 1 + 1.94T + 5T^{2} \) |
| 7 | \( 1 + 2.33T + 7T^{2} \) |
| 13 | \( 1 - 1.20T + 13T^{2} \) |
| 17 | \( 1 - 2.45T + 17T^{2} \) |
| 19 | \( 1 - 3.98T + 19T^{2} \) |
| 23 | \( 1 + 3.96T + 23T^{2} \) |
| 29 | \( 1 - 7.09T + 29T^{2} \) |
| 31 | \( 1 + 4.33T + 31T^{2} \) |
| 37 | \( 1 - 5.37T + 37T^{2} \) |
| 41 | \( 1 + 0.658T + 41T^{2} \) |
| 43 | \( 1 - 7.52T + 43T^{2} \) |
| 47 | \( 1 - 6.40T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 + 1.28T + 59T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 + 3.13T + 71T^{2} \) |
| 73 | \( 1 + 0.981T + 73T^{2} \) |
| 79 | \( 1 + 7.85T + 79T^{2} \) |
| 83 | \( 1 + 0.543T + 83T^{2} \) |
| 89 | \( 1 + 3.63T + 89T^{2} \) |
| 97 | \( 1 + 10.7T + 97T^{2} \) |
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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
−8.830581000093094841210504230479, −7.914551800523108558230750005164, −7.37527413831979933592463075307, −6.24559006901343248498972685575, −5.62373339538964953407348031483, −4.55192555908165981086166540235, −3.89417832760755320865296878310, −3.09427569684751321414514729698, −1.13216430346626463757447886548, 0,
1.13216430346626463757447886548, 3.09427569684751321414514729698, 3.89417832760755320865296878310, 4.55192555908165981086166540235, 5.62373339538964953407348031483, 6.24559006901343248498972685575, 7.37527413831979933592463075307, 7.914551800523108558230750005164, 8.830581000093094841210504230479
