| L(s) = 1 | + 5-s + 5·7-s − 3·11-s + 4·13-s + 3·17-s − 2·19-s − 6·23-s + 25-s − 6·29-s + 8·31-s + 5·35-s − 2·37-s − 12·41-s + 43-s − 6·47-s + 18·49-s − 3·53-s − 3·55-s − 5·61-s + 4·65-s − 5·67-s − 15·71-s − 4·73-s − 15·77-s − 10·79-s + 6·83-s + 3·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.88·7-s − 0.904·11-s + 1.10·13-s + 0.727·17-s − 0.458·19-s − 1.25·23-s + 1/5·25-s − 1.11·29-s + 1.43·31-s + 0.845·35-s − 0.328·37-s − 1.87·41-s + 0.152·43-s − 0.875·47-s + 18/7·49-s − 0.412·53-s − 0.404·55-s − 0.640·61-s + 0.496·65-s − 0.610·67-s − 1.78·71-s − 0.468·73-s − 1.70·77-s − 1.12·79-s + 0.658·83-s + 0.325·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−14.33285337309502, −13.59033219316958, −13.55201137335962, −12.90800293852524, −12.10690170261947, −11.72320745923481, −11.39142040731647, −10.60423666535574, −10.42924535016299, −9.958943078510845, −9.066598419674379, −8.598334785950662, −8.127271358330031, −7.822196133882879, −7.279819788919896, −6.383151649109335, −5.932002735453706, −5.405642516723461, −4.861183332141693, −4.418538681059690, −3.677249792388824, −3.006676382866707, −2.132953959695213, −1.686982333560349, −1.165889502576393, 0,
1.165889502576393, 1.686982333560349, 2.132953959695213, 3.006676382866707, 3.677249792388824, 4.418538681059690, 4.861183332141693, 5.405642516723461, 5.932002735453706, 6.383151649109335, 7.279819788919896, 7.822196133882879, 8.127271358330031, 8.598334785950662, 9.066598419674379, 9.958943078510845, 10.42924535016299, 10.60423666535574, 11.39142040731647, 11.72320745923481, 12.10690170261947, 12.90800293852524, 13.55201137335962, 13.59033219316958, 14.33285337309502
