| L(s) = 1 | + 18.0·5-s + 0.213·7-s + 3.39·11-s − 90.7·13-s + 2.59·17-s + 19·19-s − 26.6·23-s + 201.·25-s − 60.1·29-s − 176.·31-s + 3.85·35-s − 154.·37-s − 434.·41-s − 365.·43-s − 204.·47-s − 342.·49-s + 135.·53-s + 61.3·55-s − 759.·59-s + 284.·61-s − 1.63e3·65-s + 590.·67-s + 972.·71-s + 368.·73-s + 0.724·77-s + 204.·79-s + 782.·83-s + ⋯ |
| L(s) = 1 | + 1.61·5-s + 0.0115·7-s + 0.0930·11-s − 1.93·13-s + 0.0370·17-s + 0.229·19-s − 0.241·23-s + 1.61·25-s − 0.384·29-s − 1.02·31-s + 0.0186·35-s − 0.684·37-s − 1.65·41-s − 1.29·43-s − 0.633·47-s − 0.999·49-s + 0.351·53-s + 0.150·55-s − 1.67·59-s + 0.598·61-s − 3.12·65-s + 1.07·67-s + 1.62·71-s + 0.590·73-s + 0.00107·77-s + 0.291·79-s + 1.03·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - 19T \) |
| good | 5 | \( 1 - 18.0T + 125T^{2} \) |
| 7 | \( 1 - 0.213T + 343T^{2} \) |
| 11 | \( 1 - 3.39T + 1.33e3T^{2} \) |
| 13 | \( 1 + 90.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 2.59T + 4.91e3T^{2} \) |
| 23 | \( 1 + 26.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 60.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 176.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 154.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 434.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 365.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 204.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 135.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 759.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 284.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 590.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 972.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 368.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 204.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 782.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 213.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.21e3T + 9.12e5T^{2} \) |
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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
−9.068647489131669600249610655651, −8.009009933718230402219745236064, −7.01688647265585237993512389741, −6.39922484720330460268081241548, −5.23325723089623603596307916115, −5.02080174114826269837897050614, −3.42140572019635832324550938963, −2.30710041995329287223382689693, −1.65841449312822812897192127450, 0,
1.65841449312822812897192127450, 2.30710041995329287223382689693, 3.42140572019635832324550938963, 5.02080174114826269837897050614, 5.23325723089623603596307916115, 6.39922484720330460268081241548, 7.01688647265585237993512389741, 8.009009933718230402219745236064, 9.068647489131669600249610655651
