| L(s) = 1 | + 1.87·2-s − 3-s + 2.53·4-s − 1.87·6-s + 1.53·7-s + 2.87·8-s + 9-s − 0.347·11-s − 2.53·12-s − 1.87·13-s + 2.87·14-s + 2.87·16-s − 0.347·17-s + 1.87·18-s − 1.53·21-s − 0.652·22-s − 2.87·24-s − 3.53·26-s − 27-s + 3.87·28-s − 29-s + 2.53·32-s + 0.347·33-s − 0.652·34-s + 2.53·36-s + 1.87·39-s + 41-s + ⋯ |
| L(s) = 1 | + 1.87·2-s − 3-s + 2.53·4-s − 1.87·6-s + 1.53·7-s + 2.87·8-s + 9-s − 0.347·11-s − 2.53·12-s − 1.87·13-s + 2.87·14-s + 2.87·16-s − 0.347·17-s + 1.87·18-s − 1.53·21-s − 0.652·22-s − 2.87·24-s − 3.53·26-s − 27-s + 3.87·28-s − 29-s + 2.53·32-s + 0.347·33-s − 0.652·34-s + 2.53·36-s + 1.87·39-s + 41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.957401297\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.957401297\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 1.87T + T^{2} \) |
| 7 | \( 1 - 1.53T + T^{2} \) |
| 11 | \( 1 + 0.347T + T^{2} \) |
| 13 | \( 1 + 1.87T + T^{2} \) |
| 17 | \( 1 + 0.347T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + 1.53T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 0.347T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 1.87T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.523391246911277236197277394714, −7.898304037184150965892262877441, −7.43982266164931105211698921052, −6.68578315958118128747274737475, −5.72934784548585606706766407868, −4.99712873200408410429398761002, −4.80202232634207885304732900559, −3.92581744283433777952756655802, −2.51610797159669187321861614201, −1.73070337497445164535221643354,
1.73070337497445164535221643354, 2.51610797159669187321861614201, 3.92581744283433777952756655802, 4.80202232634207885304732900559, 4.99712873200408410429398761002, 5.72934784548585606706766407868, 6.68578315958118128747274737475, 7.43982266164931105211698921052, 7.898304037184150965892262877441, 9.523391246911277236197277394714
