| L(s) = 1 | + 1.23i·3-s + 1.23i·5-s − 7-s + 1.47·9-s + 4i·11-s + 1.23i·13-s − 1.52·15-s − 2·17-s − 1.23i·19-s − 1.23i·21-s − 6.47·23-s + 3.47·25-s + 5.52i·27-s − 1.52i·29-s − 4.94·33-s + ⋯ |
| L(s) = 1 | + 0.713i·3-s + 0.552i·5-s − 0.377·7-s + 0.490·9-s + 1.20i·11-s + 0.342i·13-s − 0.394·15-s − 0.485·17-s − 0.283i·19-s − 0.269i·21-s − 1.34·23-s + 0.694·25-s + 1.06i·27-s − 0.283i·29-s − 0.860·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.477769 + 1.15343i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.477769 + 1.15343i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 - 1.23iT - 3T^{2} \) |
| 5 | \( 1 - 1.23iT - 5T^{2} \) |
| 11 | \( 1 - 4iT - 11T^{2} \) |
| 13 | \( 1 - 1.23iT - 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 1.23iT - 19T^{2} \) |
| 23 | \( 1 + 6.47T + 23T^{2} \) |
| 29 | \( 1 + 1.52iT - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 6.47iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 8.94iT - 43T^{2} \) |
| 47 | \( 1 + 12.9T + 47T^{2} \) |
| 53 | \( 1 + 8.94iT - 53T^{2} \) |
| 59 | \( 1 - 9.23iT - 59T^{2} \) |
| 61 | \( 1 + 1.23iT - 61T^{2} \) |
| 67 | \( 1 + 1.52iT - 67T^{2} \) |
| 71 | \( 1 - 4.94T + 71T^{2} \) |
| 73 | \( 1 + 14.9T + 73T^{2} \) |
| 79 | \( 1 - 4.94T + 79T^{2} \) |
| 83 | \( 1 + 9.23iT - 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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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
−10.08094021964655364700250024315, −9.919802733009449185171922806969, −8.980246102714304525752714558818, −7.82954252967168014599110119930, −6.92691131438835846909974727840, −6.29386584014405064527003222456, −4.85795408176106298250936569678, −4.27599184651814684751458922963, −3.15941686988718938416533621552, −1.88438909290980683443465420867,
0.59135158652485630825189316307, 1.92594269324303370495350120388, 3.31391810475928555357680337962, 4.38465991985311912590242925759, 5.60677564396887907276409910943, 6.33724514388395487561318906033, 7.25869379194106213832712548659, 8.162102796630090532421425399777, 8.796410342433046197759941094398, 9.788847558082812662799754287174
