| L(s) = 1 | − 3.23i·3-s + 2.23·5-s + 0.763i·7-s − 7.47·9-s − 7.23i·15-s + 2.47·21-s + 5.70i·23-s + 5.00·25-s + 14.4i·27-s + 6·29-s + 1.70i·35-s − 4.47·41-s − 11.2i·43-s − 16.7·45-s + 13.7i·47-s + ⋯ |
| L(s) = 1 | − 1.86i·3-s + 0.999·5-s + 0.288i·7-s − 2.49·9-s − 1.86i·15-s + 0.539·21-s + 1.19i·23-s + 1.00·25-s + 2.78i·27-s + 1.11·29-s + 0.288i·35-s − 0.698·41-s − 1.71i·43-s − 2.49·45-s + 1.99i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.876304 - 0.876304i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.876304 - 0.876304i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 2.23T \) |
| good | 3 | \( 1 + 3.23iT - 3T^{2} \) |
| 7 | \( 1 - 0.763iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 5.70iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 4.47T + 41T^{2} \) |
| 43 | \( 1 + 11.2iT - 43T^{2} \) |
| 47 | \( 1 - 13.7iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 + 8.18iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 17.7iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−12.64281248849550552870984957719, −12.02634511974664323094929292530, −10.83802470521550071890652379357, −9.376028053791290275288534376888, −8.348622798880760706844654204351, −7.24428262020010073229159694421, −6.31002618409748606745822405705, −5.43352400992124616358714228380, −2.75334185645567540086841645499, −1.49656336476859230117027825145,
2.86185531446723818518215656358, 4.32262036432335171751972215288, 5.27231567091486983092416864152, 6.43519184526866561268525595499, 8.439039713800605599546754603685, 9.300931496355539349042987080767, 10.23370231809998942513444693926, 10.64189575891606623841671700192, 11.91169353785639270521400159309, 13.39568731903387427016449487468
