| L(s) = 1 | + 2·2-s + 2·4-s + 5-s − 3·7-s + 2·10-s − 11-s − 6·14-s − 4·16-s + 17-s + 2·19-s + 2·20-s − 2·22-s + 3·23-s + 25-s − 6·28-s + 2·29-s + 6·31-s − 8·32-s + 2·34-s − 3·35-s − 11·37-s + 4·38-s − 5·41-s + 4·43-s − 2·44-s + 6·46-s − 10·47-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 0.447·5-s − 1.13·7-s + 0.632·10-s − 0.301·11-s − 1.60·14-s − 16-s + 0.242·17-s + 0.458·19-s + 0.447·20-s − 0.426·22-s + 0.625·23-s + 1/5·25-s − 1.13·28-s + 0.371·29-s + 1.07·31-s − 1.41·32-s + 0.342·34-s − 0.507·35-s − 1.80·37-s + 0.648·38-s − 0.780·41-s + 0.609·43-s − 0.301·44-s + 0.884·46-s − 1.45·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 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
−6.97288237598139248105170679910, −6.78730601768266828130824323705, −5.94956741511883122450312015895, −5.40308420944262869382617364704, −4.76920163159057665883560579516, −3.93147197184164067152944908506, −3.03524934485667323143164240458, −2.85640501295444381229018927641, −1.54394940274433662767522970828, 0,
1.54394940274433662767522970828, 2.85640501295444381229018927641, 3.03524934485667323143164240458, 3.93147197184164067152944908506, 4.76920163159057665883560579516, 5.40308420944262869382617364704, 5.94956741511883122450312015895, 6.78730601768266828130824323705, 6.97288237598139248105170679910
