| L(s) = 1 | + 5.65i·3-s − 7·5-s + 11·7-s − 23.0·9-s − 3·11-s − 11.3i·13-s − 39.5i·15-s − 17·17-s + 19·19-s + 62.2i·21-s + 2·23-s + 24·25-s − 79.1i·27-s − 39.5i·29-s − 5.65i·31-s + ⋯ |
| L(s) = 1 | + 1.88i·3-s − 1.40·5-s + 1.57·7-s − 2.55·9-s − 0.272·11-s − 0.870i·13-s − 2.63i·15-s − 17-s + 19-s + 2.96i·21-s + 0.0869·23-s + 0.959·25-s − 2.93i·27-s − 1.36i·29-s − 0.182i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.032903214\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.032903214\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - 19T \) |
| good | 3 | \( 1 - 5.65iT - 9T^{2} \) |
| 5 | \( 1 + 7T + 25T^{2} \) |
| 7 | \( 1 - 11T + 49T^{2} \) |
| 11 | \( 1 + 3T + 121T^{2} \) |
| 13 | \( 1 + 11.3iT - 169T^{2} \) |
| 17 | \( 1 + 17T + 289T^{2} \) |
| 23 | \( 1 - 2T + 529T^{2} \) |
| 29 | \( 1 + 39.5iT - 841T^{2} \) |
| 31 | \( 1 + 5.65iT - 961T^{2} \) |
| 37 | \( 1 + 39.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 39.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 21T + 1.84e3T^{2} \) |
| 47 | \( 1 + 5T + 2.20e3T^{2} \) |
| 53 | \( 1 - 5.65iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 33.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 23T + 3.72e3T^{2} \) |
| 67 | \( 1 - 39.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 90.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 39T + 5.32e3T^{2} \) |
| 79 | \( 1 - 96.1iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 6T + 6.88e3T^{2} \) |
| 89 | \( 1 + 118. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 169. iT - 9.40e3T^{2} \) |
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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
−9.540650767681941359692805003669, −8.696617820803290269288328945828, −8.069110672223329634107191782531, −7.48203918408387780751839318903, −5.71792507495550702804150762572, −5.00628248639374506903597141782, −4.30850473646962869543015857749, −3.74205890719660940708967061462, −2.58633157320536992599630186008, −0.36222668978196687447089528847,
1.04741419332535187604488460019, 1.89600735500105100586919915613, 3.09913168100372625707754702975, 4.47404142414167805566248661023, 5.29699890288535715207902447001, 6.61638475281878046062824794978, 7.19874399972789329117563402456, 7.892922880357853441373294662679, 8.299996811489996487599868109139, 9.062383584833459763181741976473
