| L(s) = 1 | − 2-s + 4-s − 8-s + 2·11-s + 4·13-s + 16-s − 19-s − 2·22-s − 8·23-s − 4·26-s − 2·29-s − 2·31-s − 32-s + 8·37-s + 38-s − 2·41-s − 4·43-s + 2·44-s + 8·46-s + 4·47-s − 7·49-s + 4·52-s − 2·53-s + 2·58-s − 10·61-s + 2·62-s + 64-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 0.603·11-s + 1.10·13-s + 1/4·16-s − 0.229·19-s − 0.426·22-s − 1.66·23-s − 0.784·26-s − 0.371·29-s − 0.359·31-s − 0.176·32-s + 1.31·37-s + 0.162·38-s − 0.312·41-s − 0.609·43-s + 0.301·44-s + 1.17·46-s + 0.583·47-s − 49-s + 0.554·52-s − 0.274·53-s + 0.262·58-s − 1.28·61-s + 0.254·62-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.61986472172441140029633725276, −6.72870855469494060656956789272, −6.15200373065494377052148647298, −5.66587205248938053119139014089, −4.47117862991889901547772225984, −3.84290632308267884507260342246, −3.02597703266850218057670872385, −1.95286353910041906363940110984, −1.26997720328741880473546784207, 0,
1.26997720328741880473546784207, 1.95286353910041906363940110984, 3.02597703266850218057670872385, 3.84290632308267884507260342246, 4.47117862991889901547772225984, 5.66587205248938053119139014089, 6.15200373065494377052148647298, 6.72870855469494060656956789272, 7.61986472172441140029633725276
