| L(s) = 1 | − 2-s + 3-s − 4-s + 2·5-s − 6-s + 7-s + 3·8-s + 9-s − 2·10-s + 11-s − 12-s + 4·13-s − 14-s + 2·15-s − 16-s − 17-s − 18-s + 19-s − 2·20-s + 21-s − 22-s − 9·23-s + 3·24-s − 25-s − 4·26-s + 27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.894·5-s − 0.408·6-s + 0.377·7-s + 1.06·8-s + 1/3·9-s − 0.632·10-s + 0.301·11-s − 0.288·12-s + 1.10·13-s − 0.267·14-s + 0.516·15-s − 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.229·19-s − 0.447·20-s + 0.218·21-s − 0.213·22-s − 1.87·23-s + 0.612·24-s − 1/5·25-s − 0.784·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17661 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17661 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−16.23279640196208, −15.69834271566756, −14.86591138860170, −14.29808688402055, −13.90773366683824, −13.45121913958131, −13.08000355519303, −12.26410776844592, −11.55214589808083, −10.91336792041993, −10.21739944725582, −9.897909864385391, −9.278726596317953, −8.797458340172030, −8.250592562806226, −7.797701195950336, −7.073127027090934, −6.155397652307470, −5.789416075507760, −4.884256041114560, −4.149028803934321, −3.675060000630122, −2.606408687034312, −1.658623307913411, −1.392076109631084, 0,
1.392076109631084, 1.658623307913411, 2.606408687034312, 3.675060000630122, 4.149028803934321, 4.884256041114560, 5.789416075507760, 6.155397652307470, 7.073127027090934, 7.797701195950336, 8.250592562806226, 8.797458340172030, 9.278726596317953, 9.897909864385391, 10.21739944725582, 10.91336792041993, 11.55214589808083, 12.26410776844592, 13.08000355519303, 13.45121913958131, 13.90773366683824, 14.29808688402055, 14.86591138860170, 15.69834271566756, 16.23279640196208
