| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 7-s − 8-s + 9-s − 10-s + 11-s + 12-s + 14-s + 15-s + 16-s − 18-s − 8·19-s + 20-s − 21-s − 22-s + 6·23-s − 24-s + 25-s + 27-s − 28-s + 6·29-s − 30-s + 4·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s + 0.288·12-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.235·18-s − 1.83·19-s + 0.223·20-s − 0.218·21-s − 0.213·22-s + 1.25·23-s − 0.204·24-s + 1/5·25-s + 0.192·27-s − 0.188·28-s + 1.11·29-s − 0.182·30-s + 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.909735039\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.909735039\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−12.50716017569171, −11.95287538854517, −11.55044380391326, −10.95550406822032, −10.39794263005250, −10.22862742550671, −9.796574965018555, −9.187191948663289, −8.716344741560213, −8.559755544312725, −8.116042201427379, −7.433570551039617, −6.815576813620150, −6.605577301836440, −6.297948378722853, −5.518170785334981, −4.883097181901509, −4.569354545167495, −3.732488341745704, −3.339521813529862, −2.759731183286789, −2.238418430936481, −1.730603957893459, −1.140231141040810, −0.3903581703743520,
0.3903581703743520, 1.140231141040810, 1.730603957893459, 2.238418430936481, 2.759731183286789, 3.339521813529862, 3.732488341745704, 4.569354545167495, 4.883097181901509, 5.518170785334981, 6.297948378722853, 6.605577301836440, 6.815576813620150, 7.433570551039617, 8.116042201427379, 8.559755544312725, 8.716344741560213, 9.187191948663289, 9.796574965018555, 10.22862742550671, 10.39794263005250, 10.95550406822032, 11.55044380391326, 11.95287538854517, 12.50716017569171
