| L(s) = 1 | + 11.2·3-s − 19.2·5-s + 33.9i·7-s + 45.7·9-s + (−2.03 + 120. i)11-s − 72.2i·13-s − 216.·15-s + 517. i·17-s + 411. i·19-s + 382. i·21-s + 967.·23-s − 254.·25-s − 396.·27-s − 640. i·29-s − 716.·31-s + ⋯ |
| L(s) = 1 | + 1.25·3-s − 0.770·5-s + 0.692i·7-s + 0.564·9-s + (−0.0167 + 0.999i)11-s − 0.427i·13-s − 0.963·15-s + 1.79i·17-s + 1.14i·19-s + 0.866i·21-s + 1.82·23-s − 0.406·25-s − 0.544·27-s − 0.761i·29-s − 0.745·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0167 - 0.999i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.0167 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.052620829\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.052620829\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + (2.03 - 120. i)T \) |
| good | 3 | \( 1 - 11.2T + 81T^{2} \) |
| 5 | \( 1 + 19.2T + 625T^{2} \) |
| 7 | \( 1 - 33.9iT - 2.40e3T^{2} \) |
| 13 | \( 1 + 72.2iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 517. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 411. iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 967.T + 2.79e5T^{2} \) |
| 29 | \( 1 + 640. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 716.T + 9.23e5T^{2} \) |
| 37 | \( 1 - 1.20e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 966. iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 2.72e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 177.T + 4.87e6T^{2} \) |
| 53 | \( 1 + 4.02e3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 1.98e3T + 1.21e7T^{2} \) |
| 61 | \( 1 - 2.39e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 2.00e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 4.25e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 1.28e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 1.48e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 1.01e4iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 5.13e3T + 6.27e7T^{2} \) |
| 97 | \( 1 - 2.23e3T + 8.85e7T^{2} \) |
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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
−12.52865114787149798963094615201, −11.33105758207708044305945615275, −10.08403211589838880826163166553, −9.047016428847625956627586145971, −8.172309303028481452628845598075, −7.50405042698457363415400353183, −5.90159629346529467564604498699, −4.25965013181730286989820552996, −3.19120030003018836432164258892, −1.85844272296792069921832938878,
0.65421129923708536656649908242, 2.75319924338731336957866415556, 3.63890038093974606249580929413, 4.97923637518562481977075944988, 6.96066361171237420373720509666, 7.64585889014894392290185240239, 8.813265512716154201510306838310, 9.369062176092942806076700413628, 10.97389110074375160764357886651, 11.56833360103969826122378403752
