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
−11.07319548711424601985175575109, −10.07463046766373963754660240160, −9.237143862488118948839855679287, −8.433972480414300262550424124545, −7.51094004910333304812593496727, −6.77189985001450893394108442813, −5.31684806340788835560257312017, −3.96252354675476315814028854240, −2.49297216099541138085623842689, −2.16298058156741647241577654200,
1.24982926576185400563320529431, 2.62832062862750246038081177058, 3.91367758559379156718443073041, 4.80700864287449868226447695185, 6.15027973573069381665367720898, 7.67917608411217292542482597027, 8.383128496195551604994442192688, 8.833270301828422426249403664850, 9.933365423020679643499316114051, 10.69206674783284585232727479810