| L(s) = 1 | − 2.56·2-s + 4.56·4-s − 5-s − 0.438·7-s − 6.56·8-s + 2.56·10-s + 1.56·11-s − 13-s + 1.12·14-s + 7.68·16-s − 1.56·17-s − 5.12·19-s − 4.56·20-s − 4·22-s − 2.43·23-s + 25-s + 2.56·26-s − 2·28-s + 7.12·29-s + 6·31-s − 6.56·32-s + 4·34-s + 0.438·35-s − 10.6·37-s + 13.1·38-s + 6.56·40-s − 3.56·41-s + ⋯ |
| L(s) = 1 | − 1.81·2-s + 2.28·4-s − 0.447·5-s − 0.165·7-s − 2.31·8-s + 0.810·10-s + 0.470·11-s − 0.277·13-s + 0.300·14-s + 1.92·16-s − 0.378·17-s − 1.17·19-s − 1.01·20-s − 0.852·22-s − 0.508·23-s + 0.200·25-s + 0.502·26-s − 0.377·28-s + 1.32·29-s + 1.07·31-s − 1.15·32-s + 0.685·34-s + 0.0741·35-s − 1.75·37-s + 2.12·38-s + 1.03·40-s − 0.556·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 2.56T + 2T^{2} \) |
| 7 | \( 1 + 0.438T + 7T^{2} \) |
| 11 | \( 1 - 1.56T + 11T^{2} \) |
| 17 | \( 1 + 1.56T + 17T^{2} \) |
| 19 | \( 1 + 5.12T + 19T^{2} \) |
| 23 | \( 1 + 2.43T + 23T^{2} \) |
| 29 | \( 1 - 7.12T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 + 3.56T + 41T^{2} \) |
| 43 | \( 1 - 3.12T + 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 - 4.68T + 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 + 6.68T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 4.68T + 79T^{2} \) |
| 83 | \( 1 - 16.4T + 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 - 16.9T + 97T^{2} \) |
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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
−10.25647380054202828006587558414, −9.279297698324351119466275768764, −8.536522635467047420772977550802, −7.925785096726287958543361634388, −6.83819828586910433106762194099, −6.30593074763065902148610787542, −4.54113818883496080611787069092, −2.99613946156065026333088922268, −1.66120589734537228912904795573, 0,
1.66120589734537228912904795573, 2.99613946156065026333088922268, 4.54113818883496080611787069092, 6.30593074763065902148610787542, 6.83819828586910433106762194099, 7.925785096726287958543361634388, 8.536522635467047420772977550802, 9.279297698324351119466275768764, 10.25647380054202828006587558414
