| L(s) = 1 | + 2.37·2-s + 3.63·4-s + 1.31·5-s − 0.859·7-s + 3.87·8-s + 3.12·10-s − 4.30·11-s − 4.15·13-s − 2.03·14-s + 1.93·16-s − 2.08·17-s − 2.45·19-s + 4.78·20-s − 10.2·22-s − 23-s − 3.26·25-s − 9.86·26-s − 3.12·28-s + 29-s − 4.70·31-s − 3.16·32-s − 4.95·34-s − 1.13·35-s − 3.94·37-s − 5.82·38-s + 5.10·40-s + 3.17·41-s + ⋯ |
| L(s) = 1 | + 1.67·2-s + 1.81·4-s + 0.589·5-s − 0.324·7-s + 1.37·8-s + 0.989·10-s − 1.29·11-s − 1.15·13-s − 0.545·14-s + 0.483·16-s − 0.506·17-s − 0.563·19-s + 1.07·20-s − 2.18·22-s − 0.208·23-s − 0.652·25-s − 1.93·26-s − 0.590·28-s + 0.185·29-s − 0.845·31-s − 0.558·32-s − 0.850·34-s − 0.191·35-s − 0.648·37-s − 0.945·38-s + 0.807·40-s + 0.495·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 2.37T + 2T^{2} \) |
| 5 | \( 1 - 1.31T + 5T^{2} \) |
| 7 | \( 1 + 0.859T + 7T^{2} \) |
| 11 | \( 1 + 4.30T + 11T^{2} \) |
| 13 | \( 1 + 4.15T + 13T^{2} \) |
| 17 | \( 1 + 2.08T + 17T^{2} \) |
| 19 | \( 1 + 2.45T + 19T^{2} \) |
| 31 | \( 1 + 4.70T + 31T^{2} \) |
| 37 | \( 1 + 3.94T + 37T^{2} \) |
| 41 | \( 1 - 3.17T + 41T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 - 14.0T + 53T^{2} \) |
| 59 | \( 1 + 8.47T + 59T^{2} \) |
| 61 | \( 1 + 6.39T + 61T^{2} \) |
| 67 | \( 1 - 4.74T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 - 8.54T + 73T^{2} \) |
| 79 | \( 1 + 13.0T + 79T^{2} \) |
| 83 | \( 1 - 14.2T + 83T^{2} \) |
| 89 | \( 1 + 0.795T + 89T^{2} \) |
| 97 | \( 1 - 3.92T + 97T^{2} \) |
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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
−7.44003608513204842748008512364, −6.78274179412337467027663177547, −6.05209567347928139165513316013, −5.44952454536747534186256694419, −4.90405710993385520082655908949, −4.18859262561673930065876892211, −3.28571094370469433719717481524, −2.44416736722686974584996205036, −2.04464613393461471101326556383, 0,
2.04464613393461471101326556383, 2.44416736722686974584996205036, 3.28571094370469433719717481524, 4.18859262561673930065876892211, 4.90405710993385520082655908949, 5.44952454536747534186256694419, 6.05209567347928139165513316013, 6.78274179412337467027663177547, 7.44003608513204842748008512364
