| L(s) = 1 | − 2-s − 4-s + 7-s + 3·8-s − 6·13-s − 14-s − 16-s − 2·17-s + 8·19-s + 8·23-s + 6·26-s − 28-s − 2·29-s + 4·31-s − 5·32-s + 2·34-s + 2·37-s − 8·38-s − 6·41-s + 4·43-s − 8·46-s + 8·47-s + 49-s + 6·52-s + 10·53-s + 3·56-s + 2·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.377·7-s + 1.06·8-s − 1.66·13-s − 0.267·14-s − 1/4·16-s − 0.485·17-s + 1.83·19-s + 1.66·23-s + 1.17·26-s − 0.188·28-s − 0.371·29-s + 0.718·31-s − 0.883·32-s + 0.342·34-s + 0.328·37-s − 1.29·38-s − 0.937·41-s + 0.609·43-s − 1.17·46-s + 1.16·47-s + 1/7·49-s + 0.832·52-s + 1.37·53-s + 0.400·56-s + 0.262·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.825876929\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.825876929\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−13.17568306281396, −12.60347097187665, −12.11934049255719, −11.68172158496988, −11.16513003250985, −10.68130738499555, −10.10239652995665, −9.762579051167062, −9.347154166718964, −8.845571707275328, −8.525456746643527, −7.713915194847941, −7.397297643653930, −7.202728143460093, −6.501988462912632, −5.521940394139265, −5.311418024188911, −4.696526732880632, −4.448675590202783, −3.585424404118599, −2.994499379531987, −2.387080795046817, −1.734824362380236, −0.8725433982422618, −0.5881733644808502,
0.5881733644808502, 0.8725433982422618, 1.734824362380236, 2.387080795046817, 2.994499379531987, 3.585424404118599, 4.448675590202783, 4.696526732880632, 5.311418024188911, 5.521940394139265, 6.501988462912632, 7.202728143460093, 7.397297643653930, 7.713915194847941, 8.525456746643527, 8.845571707275328, 9.347154166718964, 9.762579051167062, 10.10239652995665, 10.68130738499555, 11.16513003250985, 11.68172158496988, 12.11934049255719, 12.60347097187665, 13.17568306281396
