L(s) = 1 | + 3.21i·3-s + (−0.370 + 2.20i)5-s − 2.59i·7-s − 7.35·9-s − 0.741·11-s + 3.78i·13-s + (−7.09 − 1.19i)15-s − 3.16i·17-s − 19-s + 8.35·21-s + 0.570i·23-s + (−4.72 − 1.63i)25-s − 14.0i·27-s − 6·29-s − 5.83·31-s + ⋯ |
L(s) = 1 | + 1.85i·3-s + (−0.165 + 0.986i)5-s − 0.981i·7-s − 2.45·9-s − 0.223·11-s + 1.05i·13-s + (−1.83 − 0.307i)15-s − 0.768i·17-s − 0.229·19-s + 1.82·21-s + 0.119i·23-s + (−0.945 − 0.326i)25-s − 2.69i·27-s − 1.11·29-s − 1.04·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.165 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.165 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3952273544\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3952273544\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.370 - 2.20i)T \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 - 3.21iT - 3T^{2} \) |
| 7 | \( 1 + 2.59iT - 7T^{2} \) |
| 11 | \( 1 + 0.741T + 11T^{2} \) |
| 13 | \( 1 - 3.78iT - 13T^{2} \) |
| 17 | \( 1 + 3.16iT - 17T^{2} \) |
| 23 | \( 1 - 0.570iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 5.83T + 31T^{2} \) |
| 37 | \( 1 + 1.40iT - 37T^{2} \) |
| 41 | \( 1 + 3.83T + 41T^{2} \) |
| 43 | \( 1 + 2.59iT - 43T^{2} \) |
| 47 | \( 1 + 5.08iT - 47T^{2} \) |
| 53 | \( 1 - 0.160iT - 53T^{2} \) |
| 59 | \( 1 - 8.35T + 59T^{2} \) |
| 61 | \( 1 + 8.57T + 61T^{2} \) |
| 67 | \( 1 - 14.8iT - 67T^{2} \) |
| 71 | \( 1 + 3.64T + 71T^{2} \) |
| 73 | \( 1 - 10.8iT - 73T^{2} \) |
| 79 | \( 1 - 1.83T + 79T^{2} \) |
| 83 | \( 1 + 4.19iT - 83T^{2} \) |
| 89 | \( 1 - 16.9T + 89T^{2} \) |
| 97 | \( 1 + 3.78iT - 97T^{2} \) |
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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
−10.20662788605287064643677395164, −9.403632578761846459048400596724, −8.774452152470259112172193915312, −7.56358881480493079294666170487, −6.88801972991139422593577877458, −5.77830358636773918303426833130, −4.88209532721133665914776702354, −3.93700811544765521294189576535, −3.59522069086702136390310051234, −2.38953905862629529988943235760,
0.14969015085242532126345993419, 1.47128012928217097440209312665, 2.28789214543829397353488703055, 3.44190685932704085887045676154, 5.07653513370214331826082943817, 5.75373420334060693976051194256, 6.33480563963628683144558950430, 7.56096162972803030965076372750, 7.951518468023547503240415973139, 8.691153873591038536779017345741