| L(s) = 1 | + 1.92i·3-s + 2.16·5-s + (1.74 + 1.99i)7-s − 0.716·9-s + 5.04·11-s + 13-s + 4.17i·15-s − 0.687i·17-s − 1.09i·19-s + (−3.84 + 3.35i)21-s + 3.80i·23-s − 0.304·25-s + 4.40i·27-s − 6.62i·29-s + 8.65·31-s + ⋯ |
| L(s) = 1 | + 1.11i·3-s + 0.969·5-s + (0.657 + 0.753i)7-s − 0.238·9-s + 1.52·11-s + 0.277·13-s + 1.07i·15-s − 0.166i·17-s − 0.251i·19-s + (−0.838 + 0.732i)21-s + 0.792i·23-s − 0.0609·25-s + 0.847i·27-s − 1.23i·29-s + 1.55·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.181 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.181 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.906155887\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.906155887\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-1.74 - 1.99i)T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 1.92iT - 3T^{2} \) |
| 5 | \( 1 - 2.16T + 5T^{2} \) |
| 11 | \( 1 - 5.04T + 11T^{2} \) |
| 17 | \( 1 + 0.687iT - 17T^{2} \) |
| 19 | \( 1 + 1.09iT - 19T^{2} \) |
| 23 | \( 1 - 3.80iT - 23T^{2} \) |
| 29 | \( 1 + 6.62iT - 29T^{2} \) |
| 31 | \( 1 - 8.65T + 31T^{2} \) |
| 37 | \( 1 + 9.25iT - 37T^{2} \) |
| 41 | \( 1 - 6.30iT - 41T^{2} \) |
| 43 | \( 1 - 3.27T + 43T^{2} \) |
| 47 | \( 1 + 2.09T + 47T^{2} \) |
| 53 | \( 1 + 2.89iT - 53T^{2} \) |
| 59 | \( 1 + 11.6iT - 59T^{2} \) |
| 61 | \( 1 - 1.96T + 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 + 5.07iT - 71T^{2} \) |
| 73 | \( 1 + 2.08iT - 73T^{2} \) |
| 79 | \( 1 - 0.772iT - 79T^{2} \) |
| 83 | \( 1 + 5.41iT - 83T^{2} \) |
| 89 | \( 1 + 3.65iT - 89T^{2} \) |
| 97 | \( 1 + 2.48iT - 97T^{2} \) |
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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
−9.199725266371984607491518145318, −8.455406971981387855579368857523, −7.47232820443461509693765888585, −6.31168297297140757001785754350, −5.90604671618164023213440113882, −4.95143041303437441752475119655, −4.32619151712032839901247322997, −3.45901277530171628401940257682, −2.27786709346400061330968290290, −1.34960212132479219140588852984,
1.18335308982869238172222251743, 1.45537293833609893410189917086, 2.59098079473050667894500879190, 3.90594506883258774242401514654, 4.65504216746329210998360774848, 5.78878756197663597324858009757, 6.54778747169673054500742162098, 6.86596309032249318533336989144, 7.78991818153931312411808642250, 8.518783284744084780692228854806
