| L(s) = 1 | − 2-s + 4-s + 3·5-s − 8-s − 3·10-s + 2·11-s − 2·13-s + 16-s + 17-s + 2·19-s + 3·20-s − 2·22-s + 9·23-s + 4·25-s + 2·26-s + 4·29-s − 6·31-s − 32-s − 34-s − 5·37-s − 2·38-s − 3·40-s + 5·43-s + 2·44-s − 9·46-s + 2·47-s − 7·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.34·5-s − 0.353·8-s − 0.948·10-s + 0.603·11-s − 0.554·13-s + 1/4·16-s + 0.242·17-s + 0.458·19-s + 0.670·20-s − 0.426·22-s + 1.87·23-s + 4/5·25-s + 0.392·26-s + 0.742·29-s − 1.07·31-s − 0.176·32-s − 0.171·34-s − 0.821·37-s − 0.324·38-s − 0.474·40-s + 0.762·43-s + 0.301·44-s − 1.32·46-s + 0.291·47-s − 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.577115504\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.577115504\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 61 | \( 1 - T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−9.542279411507614374384982070481, −9.362442253243670508951667136242, −8.411971412452896244672142790463, −7.25881902070036970326960022120, −6.65307190336527170048084506545, −5.68629506076367166520916726918, −4.91325264071753767766528719125, −3.31907711339124406598263296359, −2.23853402450067221315742836156, −1.15628973667305275952816214845,
1.15628973667305275952816214845, 2.23853402450067221315742836156, 3.31907711339124406598263296359, 4.91325264071753767766528719125, 5.68629506076367166520916726918, 6.65307190336527170048084506545, 7.25881902070036970326960022120, 8.411971412452896244672142790463, 9.362442253243670508951667136242, 9.542279411507614374384982070481
