| L(s) = 1 | + 0.874i·3-s + 3.70·5-s + (−1.41 − 2.23i)7-s + 2.23·9-s + 3.23·11-s + 0.874·13-s + 3.23i·15-s − 4.57i·17-s − 1.95i·19-s + (1.95 − 1.23i)21-s − 1.23i·23-s + 8.70·25-s + 4.57i·27-s + 2i·29-s − 10.2·31-s + ⋯ |
| L(s) = 1 | + 0.504i·3-s + 1.65·5-s + (−0.534 − 0.845i)7-s + 0.745·9-s + 0.975·11-s + 0.242·13-s + 0.835i·15-s − 1.10i·17-s − 0.448i·19-s + (0.426 − 0.269i)21-s − 0.257i·23-s + 1.74·25-s + 0.880i·27-s + 0.371i·29-s − 1.83·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.219i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 + 0.219i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.538808037\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.538808037\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (1.41 + 2.23i)T \) |
| good | 3 | \( 1 - 0.874iT - 3T^{2} \) |
| 5 | \( 1 - 3.70T + 5T^{2} \) |
| 11 | \( 1 - 3.23T + 11T^{2} \) |
| 13 | \( 1 - 0.874T + 13T^{2} \) |
| 17 | \( 1 + 4.57iT - 17T^{2} \) |
| 19 | \( 1 + 1.95iT - 19T^{2} \) |
| 23 | \( 1 + 1.23iT - 23T^{2} \) |
| 29 | \( 1 - 2iT - 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 + 10.9iT - 37T^{2} \) |
| 41 | \( 1 - 3.90iT - 41T^{2} \) |
| 43 | \( 1 - 1.70T + 43T^{2} \) |
| 47 | \( 1 - 8.07T + 47T^{2} \) |
| 53 | \( 1 + 0.472iT - 53T^{2} \) |
| 59 | \( 1 + 11.1iT - 59T^{2} \) |
| 61 | \( 1 + 8.94T + 61T^{2} \) |
| 67 | \( 1 + 9.70T + 67T^{2} \) |
| 71 | \( 1 + 10iT - 71T^{2} \) |
| 73 | \( 1 - 14.1iT - 73T^{2} \) |
| 79 | \( 1 - 12.4iT - 79T^{2} \) |
| 83 | \( 1 - 12.1iT - 83T^{2} \) |
| 89 | \( 1 - 9.82iT - 89T^{2} \) |
| 97 | \( 1 - 4.57iT - 97T^{2} \) |
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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
−9.419714569019436935163030222862, −8.961135812249581825304415761954, −7.33952397747941412225349268059, −6.89506329739874584127847380913, −6.04192924173010743578384714306, −5.20804899001990302489579562601, −4.27499334030728627246244858523, −3.40487643327039964204395655511, −2.14434930954531980127187770227, −1.05606400934635726525198520619,
1.49968759602565959965314114890, 1.94873296647106063895582524594, 3.21683267768335035234047326564, 4.36636158235451353594238744863, 5.69049802174982037155173669118, 6.04209733782854353011605348107, 6.68975833025831320236943166459, 7.62221370562111001047755542603, 8.961542570970635663800762346050, 9.076268968010577498459938420204
