| L(s) = 1 | + 3-s + 5-s + 7-s − 2·9-s − 3·11-s + 15-s − 3·17-s − 5·19-s + 21-s + 9·23-s + 25-s − 5·27-s − 9·29-s − 8·31-s − 3·33-s + 35-s + 7·37-s − 3·41-s − 43-s − 2·45-s − 6·49-s − 3·51-s + 6·53-s − 3·55-s − 5·57-s − 9·59-s − 61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.377·7-s − 2/3·9-s − 0.904·11-s + 0.258·15-s − 0.727·17-s − 1.14·19-s + 0.218·21-s + 1.87·23-s + 1/5·25-s − 0.962·27-s − 1.67·29-s − 1.43·31-s − 0.522·33-s + 0.169·35-s + 1.15·37-s − 0.468·41-s − 0.152·43-s − 0.298·45-s − 6/7·49-s − 0.420·51-s + 0.824·53-s − 0.404·55-s − 0.662·57-s − 1.17·59-s − 0.128·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−8.294167744689612160953219268944, −7.60757322588473593410243954858, −6.84469943386060810528121144721, −5.86288535521223664989873065993, −5.25511784388449709238563954834, −4.39734458848774032252717107221, −3.31037755216008297415525778188, −2.52066163535597727617330327108, −1.74658522130458408068637361407, 0,
1.74658522130458408068637361407, 2.52066163535597727617330327108, 3.31037755216008297415525778188, 4.39734458848774032252717107221, 5.25511784388449709238563954834, 5.86288535521223664989873065993, 6.84469943386060810528121144721, 7.60757322588473593410243954858, 8.294167744689612160953219268944
