| L(s) = 1 | + i·3-s + (1.39 + 1.74i)5-s + 3.23i·7-s − 9-s + 4.31·11-s + 2.75i·13-s + (−1.74 + 1.39i)15-s + 4.70i·17-s + 5.97·19-s − 3.23·21-s + 2.71i·23-s + (−1.11 + 4.87i)25-s − i·27-s + 5.01·29-s + 1.07·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (0.623 + 0.781i)5-s + 1.22i·7-s − 0.333·9-s + 1.30·11-s + 0.764i·13-s + (−0.451 + 0.359i)15-s + 1.14i·17-s + 1.37·19-s − 0.705·21-s + 0.565i·23-s + (−0.223 + 0.974i)25-s − 0.192i·27-s + 0.930·29-s + 0.192·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.623 - 0.781i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.623 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.546374290\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.546374290\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (-1.39 - 1.74i)T \) |
| 67 | \( 1 + iT \) |
| good | 7 | \( 1 - 3.23iT - 7T^{2} \) |
| 11 | \( 1 - 4.31T + 11T^{2} \) |
| 13 | \( 1 - 2.75iT - 13T^{2} \) |
| 17 | \( 1 - 4.70iT - 17T^{2} \) |
| 19 | \( 1 - 5.97T + 19T^{2} \) |
| 23 | \( 1 - 2.71iT - 23T^{2} \) |
| 29 | \( 1 - 5.01T + 29T^{2} \) |
| 31 | \( 1 - 1.07T + 31T^{2} \) |
| 37 | \( 1 + 6.96iT - 37T^{2} \) |
| 41 | \( 1 - 5.28T + 41T^{2} \) |
| 43 | \( 1 + 1.43iT - 43T^{2} \) |
| 47 | \( 1 + 1.42iT - 47T^{2} \) |
| 53 | \( 1 + 5.30iT - 53T^{2} \) |
| 59 | \( 1 + 2.08T + 59T^{2} \) |
| 61 | \( 1 + 3.20T + 61T^{2} \) |
| 71 | \( 1 - 1.95T + 71T^{2} \) |
| 73 | \( 1 + 2.15iT - 73T^{2} \) |
| 79 | \( 1 - 12.8T + 79T^{2} \) |
| 83 | \( 1 + 2.36iT - 83T^{2} \) |
| 89 | \( 1 - 11.6T + 89T^{2} \) |
| 97 | \( 1 + 8.77iT - 97T^{2} \) |
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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
−9.141408131812714142046397820863, −8.079136660442441832427281212785, −7.09688636544654863547692272966, −6.31884707775631955290723717499, −5.85736542997442522419679887475, −5.09027295018464355312541394334, −4.00128404847195995049368316881, −3.32315255105764323617242189450, −2.36138763923244066264413897937, −1.49968109053259515531068658198,
0.943503680214347825789437930564, 1.07871344029487663128715344588, 2.56403015277285972676655276681, 3.51080235971077508082180351246, 4.52443845396204019826429190430, 5.11438720305167895288082613229, 6.12118747418440323601631340339, 6.70160503241462636196996679389, 7.47555098736683339362630178799, 8.059847674591149315949993753533
