| L(s) = 1 | + 1.92·2-s − 3.32·3-s + 1.69·4-s + 5-s − 6.38·6-s + 2.35·7-s − 0.594·8-s + 8.04·9-s + 1.92·10-s − 11-s − 5.61·12-s − 0.122·13-s + 4.51·14-s − 3.32·15-s − 4.52·16-s − 6.40·17-s + 15.4·18-s + 4.96·19-s + 1.69·20-s − 7.81·21-s − 1.92·22-s − 2.37·23-s + 1.97·24-s + 25-s − 0.236·26-s − 16.7·27-s + 3.97·28-s + ⋯ |
| L(s) = 1 | + 1.35·2-s − 1.91·3-s + 0.845·4-s + 0.447·5-s − 2.60·6-s + 0.889·7-s − 0.210·8-s + 2.68·9-s + 0.607·10-s − 0.301·11-s − 1.62·12-s − 0.0341·13-s + 1.20·14-s − 0.858·15-s − 1.13·16-s − 1.55·17-s + 3.64·18-s + 1.13·19-s + 0.378·20-s − 1.70·21-s − 0.409·22-s − 0.494·23-s + 0.403·24-s + 0.200·25-s − 0.0463·26-s − 3.22·27-s + 0.751·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 - T \) |
| good | 2 | \( 1 - 1.92T + 2T^{2} \) |
| 3 | \( 1 + 3.32T + 3T^{2} \) |
| 7 | \( 1 - 2.35T + 7T^{2} \) |
| 13 | \( 1 + 0.122T + 13T^{2} \) |
| 17 | \( 1 + 6.40T + 17T^{2} \) |
| 19 | \( 1 - 4.96T + 19T^{2} \) |
| 23 | \( 1 + 2.37T + 23T^{2} \) |
| 29 | \( 1 + 10.3T + 29T^{2} \) |
| 31 | \( 1 + 4.08T + 31T^{2} \) |
| 37 | \( 1 - 2.13T + 37T^{2} \) |
| 41 | \( 1 - 8.90T + 41T^{2} \) |
| 43 | \( 1 - 9.77T + 43T^{2} \) |
| 47 | \( 1 - 8.00T + 47T^{2} \) |
| 53 | \( 1 - 9.34T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 + 5.82T + 61T^{2} \) |
| 67 | \( 1 - 1.67T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 79 | \( 1 + 13.0T + 79T^{2} \) |
| 83 | \( 1 + 6.38T + 83T^{2} \) |
| 89 | \( 1 + 16.5T + 89T^{2} \) |
| 97 | \( 1 + 12.0T + 97T^{2} \) |
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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
−7.50314574108980958491351692092, −7.13919773291290071843811051436, −6.06101049954481408428410361624, −5.72979969114814997604401394220, −5.22519560410668691898008551171, −4.39685069755335654171010590262, −4.05719777575320707337211984544, −2.48057610616942008352939961639, −1.46763471553236051820837834745, 0,
1.46763471553236051820837834745, 2.48057610616942008352939961639, 4.05719777575320707337211984544, 4.39685069755335654171010590262, 5.22519560410668691898008551171, 5.72979969114814997604401394220, 6.06101049954481408428410361624, 7.13919773291290071843811051436, 7.50314574108980958491351692092
