| L(s) = 1 | − 2·2-s + 9·3-s + 4·4-s − 18·6-s − 2·7-s − 8·8-s + 54·9-s − 52·11-s + 36·12-s − 43·13-s + 4·14-s + 16·16-s + 50·17-s − 108·18-s − 74·19-s − 18·21-s + 104·22-s + 23·23-s − 72·24-s + 86·26-s + 243·27-s − 8·28-s − 7·29-s − 273·31-s − 32·32-s − 468·33-s − 100·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.73·3-s + 1/2·4-s − 1.22·6-s − 0.107·7-s − 0.353·8-s + 2·9-s − 1.42·11-s + 0.866·12-s − 0.917·13-s + 0.0763·14-s + 1/4·16-s + 0.713·17-s − 1.41·18-s − 0.893·19-s − 0.187·21-s + 1.00·22-s + 0.208·23-s − 0.612·24-s + 0.648·26-s + 1.73·27-s − 0.0539·28-s − 0.0448·29-s − 1.58·31-s − 0.176·32-s − 2.46·33-s − 0.504·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p T \) |
| 5 | \( 1 \) |
| 23 | \( 1 - p T \) |
| good | 3 | \( 1 - p^{2} T + p^{3} T^{2} \) |
| 7 | \( 1 + 2 T + p^{3} T^{2} \) |
| 11 | \( 1 + 52 T + p^{3} T^{2} \) |
| 13 | \( 1 + 43 T + p^{3} T^{2} \) |
| 17 | \( 1 - 50 T + p^{3} T^{2} \) |
| 19 | \( 1 + 74 T + p^{3} T^{2} \) |
| 29 | \( 1 + 7 T + p^{3} T^{2} \) |
| 31 | \( 1 + 273 T + p^{3} T^{2} \) |
| 37 | \( 1 - 4 T + p^{3} T^{2} \) |
| 41 | \( 1 - 3 p T + p^{3} T^{2} \) |
| 43 | \( 1 - 152 T + p^{3} T^{2} \) |
| 47 | \( 1 + 75 T + p^{3} T^{2} \) |
| 53 | \( 1 + 86 T + p^{3} T^{2} \) |
| 59 | \( 1 + 444 T + p^{3} T^{2} \) |
| 61 | \( 1 - 262 T + p^{3} T^{2} \) |
| 67 | \( 1 + 764 T + p^{3} T^{2} \) |
| 71 | \( 1 + 21 T + p^{3} T^{2} \) |
| 73 | \( 1 + 681 T + p^{3} T^{2} \) |
| 79 | \( 1 - 426 T + p^{3} T^{2} \) |
| 83 | \( 1 + 902 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1272 T + p^{3} T^{2} \) |
| 97 | \( 1 - 342 T + p^{3} T^{2} \) |
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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
−8.986679506328702732379939454852, −8.185177389290577062966853076107, −7.64231911256658416147143118592, −7.06098624624126683002822184907, −5.64976082816020065105892806714, −4.46568411950519593660793908713, −3.24386534231923832223474833493, −2.58080769625535970709153776301, −1.71775450590236319929919184825, 0,
1.71775450590236319929919184825, 2.58080769625535970709153776301, 3.24386534231923832223474833493, 4.46568411950519593660793908713, 5.64976082816020065105892806714, 7.06098624624126683002822184907, 7.64231911256658416147143118592, 8.185177389290577062966853076107, 8.986679506328702732379939454852
