| L(s) = 1 | − 0.806i·3-s + (−1.48 − 1.67i)5-s − 2.15i·7-s + 2.35·9-s − 0.387·11-s − 0.962i·13-s + (−1.35 + 1.19i)15-s + 1.61i·17-s − 6.31·19-s − 1.73·21-s − 6.15i·23-s + (−0.612 + 4.96i)25-s − 4.31i·27-s − 2·29-s − 9.92·31-s + ⋯ |
| L(s) = 1 | − 0.465i·3-s + (−0.662 − 0.749i)5-s − 0.815i·7-s + 0.783·9-s − 0.116·11-s − 0.266i·13-s + (−0.348 + 0.308i)15-s + 0.390i·17-s − 1.44·19-s − 0.379·21-s − 1.28i·23-s + (−0.122 + 0.992i)25-s − 0.829i·27-s − 0.371·29-s − 1.78·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.363850 - 0.960774i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.363850 - 0.960774i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.48 + 1.67i)T \) |
| good | 3 | \( 1 + 0.806iT - 3T^{2} \) |
| 7 | \( 1 + 2.15iT - 7T^{2} \) |
| 11 | \( 1 + 0.387T + 11T^{2} \) |
| 13 | \( 1 + 0.962iT - 13T^{2} \) |
| 17 | \( 1 - 1.61iT - 17T^{2} \) |
| 19 | \( 1 + 6.31T + 19T^{2} \) |
| 23 | \( 1 + 6.15iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 9.92T + 31T^{2} \) |
| 37 | \( 1 + 6.57iT - 37T^{2} \) |
| 41 | \( 1 - 4.57T + 41T^{2} \) |
| 43 | \( 1 + 11.5iT - 43T^{2} \) |
| 47 | \( 1 - 4.54iT - 47T^{2} \) |
| 53 | \( 1 - 8.96iT - 53T^{2} \) |
| 59 | \( 1 - 6.31T + 59T^{2} \) |
| 61 | \( 1 - 0.261T + 61T^{2} \) |
| 67 | \( 1 - 9.89iT - 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 + 13.0iT - 73T^{2} \) |
| 79 | \( 1 + 1.92T + 79T^{2} \) |
| 83 | \( 1 + 2.88iT - 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 - 9.61iT - 97T^{2} \) |
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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
−10.51139105263399323137518548825, −9.226814885442788513723859142927, −8.412237946063218968572319087020, −7.52873425432985526332272893076, −6.93086086737504767099316288499, −5.70419531325168523817703881243, −4.38848025002003117857841614802, −3.88574069560824373859677896090, −2.00208103726861294802927635517, −0.55351058273099113143159957357,
2.06863485004794163643278776875, 3.39630476533070444440078717580, 4.28060099177906639086947583016, 5.39111725514372905128772070435, 6.54531321789428489611513867533, 7.34986814219781701425467321554, 8.296054167908762028417116058440, 9.297522674512721725512641865025, 10.01415236132315578584481054733, 11.02292800078921307153179024592
