| L(s) = 1 | − 0.0300·2-s + 3-s − 1.99·4-s − 0.348·5-s − 0.0300·6-s − 0.368·7-s + 0.120·8-s + 9-s + 0.0104·10-s + 5.54·11-s − 1.99·12-s + 13-s + 0.0110·14-s − 0.348·15-s + 3.99·16-s + 5.74·17-s − 0.0300·18-s − 2.87·19-s + 0.697·20-s − 0.368·21-s − 0.166·22-s − 5.41·23-s + 0.120·24-s − 4.87·25-s − 0.0300·26-s + 27-s + 0.737·28-s + ⋯ |
| L(s) = 1 | − 0.0212·2-s + 0.577·3-s − 0.999·4-s − 0.156·5-s − 0.0122·6-s − 0.139·7-s + 0.0424·8-s + 0.333·9-s + 0.00331·10-s + 1.67·11-s − 0.577·12-s + 0.277·13-s + 0.00295·14-s − 0.0900·15-s + 0.998·16-s + 1.39·17-s − 0.00707·18-s − 0.659·19-s + 0.155·20-s − 0.0804·21-s − 0.0354·22-s − 1.12·23-s + 0.0244·24-s − 0.975·25-s − 0.00588·26-s + 0.192·27-s + 0.139·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.965797488\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.965797488\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 + 0.0300T + 2T^{2} \) |
| 5 | \( 1 + 0.348T + 5T^{2} \) |
| 7 | \( 1 + 0.368T + 7T^{2} \) |
| 11 | \( 1 - 5.54T + 11T^{2} \) |
| 17 | \( 1 - 5.74T + 17T^{2} \) |
| 19 | \( 1 + 2.87T + 19T^{2} \) |
| 23 | \( 1 + 5.41T + 23T^{2} \) |
| 29 | \( 1 + 2.97T + 29T^{2} \) |
| 31 | \( 1 - 3.33T + 31T^{2} \) |
| 37 | \( 1 + 0.610T + 37T^{2} \) |
| 41 | \( 1 - 8.81T + 41T^{2} \) |
| 43 | \( 1 - 3.34T + 43T^{2} \) |
| 47 | \( 1 + 1.21T + 47T^{2} \) |
| 53 | \( 1 - 14.2T + 53T^{2} \) |
| 59 | \( 1 + 0.0252T + 59T^{2} \) |
| 61 | \( 1 - 8.94T + 61T^{2} \) |
| 67 | \( 1 + 3.03T + 67T^{2} \) |
| 71 | \( 1 - 8.52T + 71T^{2} \) |
| 73 | \( 1 + 9.89T + 73T^{2} \) |
| 79 | \( 1 + 3.87T + 79T^{2} \) |
| 83 | \( 1 + 0.105T + 83T^{2} \) |
| 89 | \( 1 + 9.35T + 89T^{2} \) |
| 97 | \( 1 + 15.4T + 97T^{2} \) |
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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
−8.470971742362937202876256528196, −7.927422441160754992211528821011, −7.11999430072214258584781072002, −6.10890992767220093957339266265, −5.57902757444903669222528699138, −4.20385499575968817623886848004, −4.05097608282070157480160041668, −3.21734124309556735242899036068, −1.85301564714444639336217089989, −0.829837513276370830023866781571,
0.829837513276370830023866781571, 1.85301564714444639336217089989, 3.21734124309556735242899036068, 4.05097608282070157480160041668, 4.20385499575968817623886848004, 5.57902757444903669222528699138, 6.10890992767220093957339266265, 7.11999430072214258584781072002, 7.927422441160754992211528821011, 8.470971742362937202876256528196
