| L(s) = 1 | − 3.95·2-s − 3·3-s + 7.65·4-s + 8.88·5-s + 11.8·6-s − 1.22·7-s + 1.36·8-s + 9·9-s − 35.1·10-s − 11·11-s − 22.9·12-s + 57.8·13-s + 4.82·14-s − 26.6·15-s − 66.6·16-s + 37.1·17-s − 35.6·18-s + 150.·19-s + 68.0·20-s + 3.66·21-s + 43.5·22-s − 96.0·23-s − 4.10·24-s − 46.0·25-s − 229.·26-s − 27·27-s − 9.34·28-s + ⋯ |
| L(s) = 1 | − 1.39·2-s − 0.577·3-s + 0.956·4-s + 0.794·5-s + 0.807·6-s − 0.0658·7-s + 0.0604·8-s + 0.333·9-s − 1.11·10-s − 0.301·11-s − 0.552·12-s + 1.23·13-s + 0.0921·14-s − 0.458·15-s − 1.04·16-s + 0.529·17-s − 0.466·18-s + 1.81·19-s + 0.760·20-s + 0.0380·21-s + 0.421·22-s − 0.871·23-s − 0.0348·24-s − 0.368·25-s − 1.72·26-s − 0.192·27-s − 0.0630·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.189879065\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.189879065\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 11 | \( 1 + 11T \) |
| 61 | \( 1 + 61T \) |
| good | 2 | \( 1 + 3.95T + 8T^{2} \) |
| 5 | \( 1 - 8.88T + 125T^{2} \) |
| 7 | \( 1 + 1.22T + 343T^{2} \) |
| 13 | \( 1 - 57.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 37.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 150.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 96.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 160.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 12.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 122.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 313.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 379.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 50.3T + 1.03e5T^{2} \) |
| 53 | \( 1 + 302.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 716.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 107.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 233.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 518.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 172.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.32e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 141.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 785.T + 9.12e5T^{2} \) |
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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
−8.961358815483107425323092556559, −7.985920291436407126696898916597, −7.54561473979036839537365172724, −6.43069822393898494812222053608, −5.87599647410817553637803535666, −4.98551510476931092710540243685, −3.77245487242997984919662930170, −2.47375223014021902177799540287, −1.34473692701098601474631013982, −0.73865272046736920133760081832,
0.73865272046736920133760081832, 1.34473692701098601474631013982, 2.47375223014021902177799540287, 3.77245487242997984919662930170, 4.98551510476931092710540243685, 5.87599647410817553637803535666, 6.43069822393898494812222053608, 7.54561473979036839537365172724, 7.985920291436407126696898916597, 8.961358815483107425323092556559
