| L(s) = 1 | + (4.42 + 0.701i)3-s + (1.95 − 4.60i)5-s + (4.77 − 4.77i)7-s + (10.5 + 3.42i)9-s + (−3.84 − 11.8i)11-s + (−1.05 + 0.536i)13-s + (11.8 − 19.0i)15-s + (−4.59 + 0.727i)17-s + (−12.8 + 17.6i)19-s + (24.4 − 17.7i)21-s + (4.03 − 7.92i)23-s + (−17.3 − 17.9i)25-s + (8.34 + 4.25i)27-s + (5.42 + 7.47i)29-s + (25.3 + 18.3i)31-s + ⋯ |
| L(s) = 1 | + (1.47 + 0.233i)3-s + (0.390 − 0.920i)5-s + (0.682 − 0.682i)7-s + (1.17 + 0.380i)9-s + (−0.349 − 1.07i)11-s + (−0.0809 + 0.0412i)13-s + (0.790 − 1.26i)15-s + (−0.270 + 0.0428i)17-s + (−0.676 + 0.931i)19-s + (1.16 − 0.847i)21-s + (0.175 − 0.344i)23-s + (−0.695 − 0.718i)25-s + (0.309 + 0.157i)27-s + (0.187 + 0.257i)29-s + (0.816 + 0.593i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 + 0.673i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.739 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.81989 - 1.09183i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.81989 - 1.09183i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.95 + 4.60i)T \) |
| good | 3 | \( 1 + (-4.42 - 0.701i)T + (8.55 + 2.78i)T^{2} \) |
| 7 | \( 1 + (-4.77 + 4.77i)T - 49iT^{2} \) |
| 11 | \( 1 + (3.84 + 11.8i)T + (-97.8 + 71.1i)T^{2} \) |
| 13 | \( 1 + (1.05 - 0.536i)T + (99.3 - 136. i)T^{2} \) |
| 17 | \( 1 + (4.59 - 0.727i)T + (274. - 89.3i)T^{2} \) |
| 19 | \( 1 + (12.8 - 17.6i)T + (-111. - 343. i)T^{2} \) |
| 23 | \( 1 + (-4.03 + 7.92i)T + (-310. - 427. i)T^{2} \) |
| 29 | \( 1 + (-5.42 - 7.47i)T + (-259. + 799. i)T^{2} \) |
| 31 | \( 1 + (-25.3 - 18.3i)T + (296. + 913. i)T^{2} \) |
| 37 | \( 1 + (-6.47 - 12.7i)T + (-804. + 1.10e3i)T^{2} \) |
| 41 | \( 1 + (16.8 - 51.9i)T + (-1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 + (-36.1 - 36.1i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-0.703 + 4.43i)T + (-2.10e3 - 682. i)T^{2} \) |
| 53 | \( 1 + (-69.2 - 10.9i)T + (2.67e3 + 868. i)T^{2} \) |
| 59 | \( 1 + (-67.9 - 22.0i)T + (2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 + (-15.0 - 46.4i)T + (-3.01e3 + 2.18e3i)T^{2} \) |
| 67 | \( 1 + (79.5 - 12.5i)T + (4.26e3 - 1.38e3i)T^{2} \) |
| 71 | \( 1 + (-34.9 + 25.3i)T + (1.55e3 - 4.79e3i)T^{2} \) |
| 73 | \( 1 + (-27.3 + 53.6i)T + (-3.13e3 - 4.31e3i)T^{2} \) |
| 79 | \( 1 + (-27.4 - 37.8i)T + (-1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (17.1 + 108. i)T + (-6.55e3 + 2.12e3i)T^{2} \) |
| 89 | \( 1 + (-63.0 + 20.4i)T + (6.40e3 - 4.65e3i)T^{2} \) |
| 97 | \( 1 + (-8.99 + 56.7i)T + (-8.94e3 - 2.90e3i)T^{2} \) |
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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
−10.69206674783284585232727479810, −9.933365423020679643499316114051, −8.833270301828422426249403664850, −8.383128496195551604994442192688, −7.67917608411217292542482597027, −6.15027973573069381665367720898, −4.80700864287449868226447695185, −3.91367758559379156718443073041, −2.62832062862750246038081177058, −1.24982926576185400563320529431,
2.16298058156741647241577654200, 2.49297216099541138085623842689, 3.96252354675476315814028854240, 5.31684806340788835560257312017, 6.77189985001450893394108442813, 7.51094004910333304812593496727, 8.433972480414300262550424124545, 9.237143862488118948839855679287, 10.07463046766373963754660240160, 11.07319548711424601985175575109
