| L(s) = 1 | + (0.618 − 1.61i)3-s + 3.23i·5-s − 1.23i·7-s + (−2.23 − 2.00i)9-s + 5.23·11-s + 4.47·13-s + (5.23 + 2.00i)15-s − 2.47i·17-s + 0.763i·19-s + (−2.00 − 0.763i)21-s + 2.47·23-s − 5.47·25-s + (−4.61 + 2.38i)27-s + 4.76i·29-s − 5.23i·31-s + ⋯ |
| L(s) = 1 | + (0.356 − 0.934i)3-s + 1.44i·5-s − 0.467i·7-s + (−0.745 − 0.666i)9-s + 1.57·11-s + 1.24·13-s + (1.35 + 0.516i)15-s − 0.599i·17-s + 0.175i·19-s + (−0.436 − 0.166i)21-s + 0.515·23-s − 1.09·25-s + (−0.888 + 0.458i)27-s + 0.884i·29-s − 0.940i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.356i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.934 + 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.60962 - 0.296948i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.60962 - 0.296948i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.618 + 1.61i)T \) |
| good | 5 | \( 1 - 3.23iT - 5T^{2} \) |
| 7 | \( 1 + 1.23iT - 7T^{2} \) |
| 11 | \( 1 - 5.23T + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 + 2.47iT - 17T^{2} \) |
| 19 | \( 1 - 0.763iT - 19T^{2} \) |
| 23 | \( 1 - 2.47T + 23T^{2} \) |
| 29 | \( 1 - 4.76iT - 29T^{2} \) |
| 31 | \( 1 + 5.23iT - 31T^{2} \) |
| 37 | \( 1 + 8.47T + 37T^{2} \) |
| 41 | \( 1 - 6.47iT - 41T^{2} \) |
| 43 | \( 1 + 7.23iT - 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 - 3.23iT - 53T^{2} \) |
| 59 | \( 1 + 1.23T + 59T^{2} \) |
| 61 | \( 1 + 0.472T + 61T^{2} \) |
| 67 | \( 1 - 9.70iT - 67T^{2} \) |
| 71 | \( 1 + 15.4T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + 0.291iT - 79T^{2} \) |
| 83 | \( 1 - 2.76T + 83T^{2} \) |
| 89 | \( 1 - 4iT - 89T^{2} \) |
| 97 | \( 1 - 0.472T + 97T^{2} \) |
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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
−11.35938087296978120152053947417, −10.56718514876331542746735153428, −9.342747798133658534827646705644, −8.460887599165852441627775716635, −7.18525770703991326931854786251, −6.79698648521432253349227155231, −5.93826331989077339506237106707, −3.86457554916008562357222693846, −3.03819750642008649545686305237, −1.45472451603873987349616811025,
1.51012220631758179412520442508, 3.52229369124652651480767571762, 4.38230861728745604736417430792, 5.38207263894208148420383547763, 6.40602062376300254341970292162, 8.160166407138318214300651917701, 8.968546524243489079159293704635, 9.110588519555404409207739862895, 10.38547358187732430650582116834, 11.45064716632547921795721179282
