| L(s) = 1 | − 2.37·3-s + 5-s + 2.62·9-s + 6.37·11-s − 4.37·13-s − 2.37·15-s + 0.372·17-s + 4.74·19-s − 4.74·23-s + 25-s + 0.883·27-s − 4.37·29-s + 8·31-s − 15.1·33-s − 2·37-s + 10.3·39-s − 6.74·41-s − 8.74·43-s + 2.62·45-s + 7.11·47-s − 0.883·51-s + 10.7·53-s + 6.37·55-s − 11.2·57-s − 8·59-s + 2.74·61-s − 4.37·65-s + ⋯ |
| L(s) = 1 | − 1.36·3-s + 0.447·5-s + 0.875·9-s + 1.92·11-s − 1.21·13-s − 0.612·15-s + 0.0902·17-s + 1.08·19-s − 0.989·23-s + 0.200·25-s + 0.169·27-s − 0.811·29-s + 1.43·31-s − 2.63·33-s − 0.328·37-s + 1.66·39-s − 1.05·41-s − 1.33·43-s + 0.391·45-s + 1.03·47-s − 0.123·51-s + 1.47·53-s + 0.859·55-s − 1.49·57-s − 1.04·59-s + 0.351·61-s − 0.542·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.179862872\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.179862872\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2.37T + 3T^{2} \) |
| 11 | \( 1 - 6.37T + 11T^{2} \) |
| 13 | \( 1 + 4.37T + 13T^{2} \) |
| 17 | \( 1 - 0.372T + 17T^{2} \) |
| 19 | \( 1 - 4.74T + 19T^{2} \) |
| 23 | \( 1 + 4.74T + 23T^{2} \) |
| 29 | \( 1 + 4.37T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 6.74T + 41T^{2} \) |
| 43 | \( 1 + 8.74T + 43T^{2} \) |
| 47 | \( 1 - 7.11T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 - 2.74T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 15.1T + 79T^{2} \) |
| 83 | \( 1 - 9.48T + 83T^{2} \) |
| 89 | \( 1 + 14.7T + 89T^{2} \) |
| 97 | \( 1 - 9.86T + 97T^{2} \) |
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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
−9.488991419004228700428521171191, −8.466197216153443266395986906478, −7.29225899201460954213517827007, −6.66863998041645263956395365037, −6.01971194581359503799376788947, −5.24403265531364040439452053680, −4.50448923309360709457113302045, −3.44813511706461029467473953842, −1.95062111430707587872792432362, −0.800343423597108698838844792882,
0.800343423597108698838844792882, 1.95062111430707587872792432362, 3.44813511706461029467473953842, 4.50448923309360709457113302045, 5.24403265531364040439452053680, 6.01971194581359503799376788947, 6.66863998041645263956395365037, 7.29225899201460954213517827007, 8.466197216153443266395986906478, 9.488991419004228700428521171191
