| L(s) = 1 | − 1.98·2-s − 0.573·3-s + 2.94·4-s + 1.13·6-s − 1.37·7-s − 3.86·8-s − 0.670·9-s − 1.68·12-s + 2.72·14-s + 4.73·16-s + 1.33·18-s + 1.19·19-s + 0.787·21-s − 0.116·23-s + 2.21·24-s + 25-s + 0.958·27-s − 4.04·28-s − 1.87·31-s − 5.53·32-s − 1.97·36-s + 1.53·37-s − 2.37·38-s + 0.347·41-s − 1.56·42-s + 0.792·43-s + 0.231·46-s + ⋯ |
| L(s) = 1 | − 1.98·2-s − 0.573·3-s + 2.94·4-s + 1.13·6-s − 1.37·7-s − 3.86·8-s − 0.670·9-s − 1.68·12-s + 2.72·14-s + 4.73·16-s + 1.33·18-s + 1.19·19-s + 0.787·21-s − 0.116·23-s + 2.21·24-s + 25-s + 0.958·27-s − 4.04·28-s − 1.87·31-s − 5.53·32-s − 1.97·36-s + 1.53·37-s − 2.37·38-s + 0.347·41-s − 1.56·42-s + 0.792·43-s + 0.231·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2600929936\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2600929936\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 983 | \( 1 - T \) |
| good | 2 | \( 1 + 1.98T + T^{2} \) |
| 3 | \( 1 + 0.573T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 + 1.37T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - 1.19T + T^{2} \) |
| 23 | \( 1 + 0.116T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + 1.87T + T^{2} \) |
| 37 | \( 1 - 1.53T + T^{2} \) |
| 41 | \( 1 - 0.347T + T^{2} \) |
| 43 | \( 1 - 0.792T + T^{2} \) |
| 47 | \( 1 - 1.78T + T^{2} \) |
| 53 | \( 1 - 0.792T + T^{2} \) |
| 59 | \( 1 + T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 0.347T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 1.94T + T^{2} \) |
| 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
−10.08205999643285150428771198376, −9.225023236244035390378918017082, −8.934136661555314395525823715075, −7.67967614716262121445043453503, −7.08197208944588354733315115370, −6.16486183679071612010962128416, −5.62953860189434050757844560299, −3.36672971197555799117340141244, −2.52750703155840666162534528489, −0.78214809640805405585048506681,
0.78214809640805405585048506681, 2.52750703155840666162534528489, 3.36672971197555799117340141244, 5.62953860189434050757844560299, 6.16486183679071612010962128416, 7.08197208944588354733315115370, 7.67967614716262121445043453503, 8.934136661555314395525823715075, 9.225023236244035390378918017082, 10.08205999643285150428771198376
