| L(s) = 1 | + 1.17·2-s + 1.80·3-s − 0.621·4-s + 5-s + 2.12·6-s − 2.77·7-s − 3.07·8-s + 0.266·9-s + 1.17·10-s − 11-s − 1.12·12-s + 4.33·13-s − 3.26·14-s + 1.80·15-s − 2.37·16-s + 4.25·17-s + 0.312·18-s − 2.81·19-s − 0.621·20-s − 5.01·21-s − 1.17·22-s − 0.836·23-s − 5.56·24-s + 25-s + 5.09·26-s − 4.94·27-s + 1.72·28-s + ⋯ |
| L(s) = 1 | + 0.830·2-s + 1.04·3-s − 0.310·4-s + 0.447·5-s + 0.866·6-s − 1.04·7-s − 1.08·8-s + 0.0887·9-s + 0.371·10-s − 0.301·11-s − 0.324·12-s + 1.20·13-s − 0.871·14-s + 0.466·15-s − 0.592·16-s + 1.03·17-s + 0.0736·18-s − 0.644·19-s − 0.138·20-s − 1.09·21-s − 0.250·22-s − 0.174·23-s − 1.13·24-s + 0.200·25-s + 0.999·26-s − 0.950·27-s + 0.326·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.436987358\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.436987358\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| good | 2 | \( 1 - 1.17T + 2T^{2} \) |
| 3 | \( 1 - 1.80T + 3T^{2} \) |
| 7 | \( 1 + 2.77T + 7T^{2} \) |
| 13 | \( 1 - 4.33T + 13T^{2} \) |
| 17 | \( 1 - 4.25T + 17T^{2} \) |
| 19 | \( 1 + 2.81T + 19T^{2} \) |
| 23 | \( 1 + 0.836T + 23T^{2} \) |
| 29 | \( 1 - 8.33T + 29T^{2} \) |
| 31 | \( 1 - 6.65T + 31T^{2} \) |
| 37 | \( 1 - 6.78T + 37T^{2} \) |
| 41 | \( 1 - 7.85T + 41T^{2} \) |
| 43 | \( 1 - 2.47T + 43T^{2} \) |
| 47 | \( 1 - 7.35T + 47T^{2} \) |
| 53 | \( 1 - 7.05T + 53T^{2} \) |
| 59 | \( 1 + 1.85T + 59T^{2} \) |
| 61 | \( 1 - 15.0T + 61T^{2} \) |
| 67 | \( 1 - 4.17T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 + 9.73T + 83T^{2} \) |
| 89 | \( 1 + 7.62T + 89T^{2} \) |
| 97 | \( 1 - 0.435T + 97T^{2} \) |
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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
−8.558431383962143730157907815112, −7.925129178630713822588306449735, −6.77080214070148835303548351109, −5.98270673038030391993050053567, −5.67303488797977281270068668816, −4.39768149419869630603497790841, −3.81682890993986834274744373256, −2.93212541754163774762372538200, −2.59967835179487612956134456029, −0.899508372282235258188201012559,
0.899508372282235258188201012559, 2.59967835179487612956134456029, 2.93212541754163774762372538200, 3.81682890993986834274744373256, 4.39768149419869630603497790841, 5.67303488797977281270068668816, 5.98270673038030391993050053567, 6.77080214070148835303548351109, 7.925129178630713822588306449735, 8.558431383962143730157907815112
