| L(s) = 1 | + 2-s − 4-s + 2·5-s − 7-s − 3·8-s + 2·10-s + 11-s − 14-s − 16-s + 4·19-s − 2·20-s + 22-s + 4·23-s − 25-s + 28-s − 2·29-s − 10·31-s + 5·32-s − 2·35-s + 4·37-s + 4·38-s − 6·40-s + 10·41-s − 2·43-s − 44-s + 4·46-s − 10·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.894·5-s − 0.377·7-s − 1.06·8-s + 0.632·10-s + 0.301·11-s − 0.267·14-s − 1/4·16-s + 0.917·19-s − 0.447·20-s + 0.213·22-s + 0.834·23-s − 1/5·25-s + 0.188·28-s − 0.371·29-s − 1.79·31-s + 0.883·32-s − 0.338·35-s + 0.657·37-s + 0.648·38-s − 0.948·40-s + 1.56·41-s − 0.304·43-s − 0.150·44-s + 0.589·46-s − 1.45·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.997774567\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.997774567\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.45943756147143, −13.19330499616526, −12.66448213448882, −12.53141720544963, −11.59910659949636, −11.31942598863964, −10.76599684789241, −9.976225409113981, −9.526691565018403, −9.370098673110201, −8.912430144428922, −8.089983293403815, −7.703844690433179, −6.824980786914985, −6.569372704129071, −5.797457554663564, −5.526174471092030, −5.057556933377244, −4.441085223600850, −3.681516959225292, −3.430282239209613, −2.679609100770244, −2.061327443750621, −1.262775709230376, −0.4917962877351000,
0.4917962877351000, 1.262775709230376, 2.061327443750621, 2.679609100770244, 3.430282239209613, 3.681516959225292, 4.441085223600850, 5.057556933377244, 5.526174471092030, 5.797457554663564, 6.569372704129071, 6.824980786914985, 7.703844690433179, 8.089983293403815, 8.912430144428922, 9.370098673110201, 9.526691565018403, 9.976225409113981, 10.76599684789241, 11.31942598863964, 11.59910659949636, 12.53141720544963, 12.66448213448882, 13.19330499616526, 13.45943756147143
