| L(s) = 1 | + 2-s + 4-s + 1.07i·5-s + 2.64·7-s + 8-s + 1.07i·10-s − 3.85·11-s + 5.91·13-s + 2.64·14-s + 16-s − 7.35i·17-s − 7.16i·19-s + 1.07i·20-s − 3.85·22-s + 1.57i·23-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.481i·5-s + 0.999·7-s + 0.353·8-s + 0.340i·10-s − 1.16·11-s + 1.63·13-s + 0.706·14-s + 0.250·16-s − 1.78i·17-s − 1.64i·19-s + 0.240i·20-s − 0.822·22-s + 0.328i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.129i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.207776049\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.207776049\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 113 | \( 1 + (10.5 + 1.38i)T \) |
| good | 5 | \( 1 - 1.07iT - 5T^{2} \) |
| 7 | \( 1 - 2.64T + 7T^{2} \) |
| 11 | \( 1 + 3.85T + 11T^{2} \) |
| 13 | \( 1 - 5.91T + 13T^{2} \) |
| 17 | \( 1 + 7.35iT - 17T^{2} \) |
| 19 | \( 1 + 7.16iT - 19T^{2} \) |
| 23 | \( 1 - 1.57iT - 23T^{2} \) |
| 29 | \( 1 + 0.967iT - 29T^{2} \) |
| 31 | \( 1 + 1.01T + 31T^{2} \) |
| 37 | \( 1 - 9.52iT - 37T^{2} \) |
| 41 | \( 1 - 3.46T + 41T^{2} \) |
| 43 | \( 1 - 6.84iT - 43T^{2} \) |
| 47 | \( 1 + 2.55iT - 47T^{2} \) |
| 53 | \( 1 - 10.5T + 53T^{2} \) |
| 59 | \( 1 - 13.8iT - 59T^{2} \) |
| 61 | \( 1 - 5.91T + 61T^{2} \) |
| 67 | \( 1 - 11.7iT - 67T^{2} \) |
| 71 | \( 1 + 3.03iT - 71T^{2} \) |
| 73 | \( 1 + 9.40iT - 73T^{2} \) |
| 79 | \( 1 + 6.08iT - 79T^{2} \) |
| 83 | \( 1 + 6.33T + 83T^{2} \) |
| 89 | \( 1 + 0.303iT - 89T^{2} \) |
| 97 | \( 1 + 12.0T + 97T^{2} \) |
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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
−8.971981908760634061712374746305, −8.288238813230687666858372263674, −7.37398092678981486448593750486, −6.83917479310810068046321379038, −5.77573519659933877281287158028, −5.02254671353505905686881929795, −4.42392709523973684555189716834, −3.09172067202817347788906542366, −2.55713441492918321199821186960, −1.07454219606424740680531417801,
1.30316443384326401879076099463, 2.15885868118422871402319694801, 3.65076767761458691414054053361, 4.12036647456626900208700831211, 5.31329371364164488156676025834, 5.69571275689026742033397731797, 6.59163503184928537204119256313, 7.87020789577745320089825803816, 8.197394294125525584409368888875, 8.865342182195044968346565771844
