| L(s) = 1 | + (1.51 − 2.98i)2-s + (−0.270 + 1.71i)3-s + (−4.23 − 5.83i)4-s + (3.83 − 3.21i)5-s + (4.69 + 3.40i)6-s + (−3.58 − 3.58i)7-s + (−10.6 + 1.68i)8-s + (−2.85 − 0.927i)9-s + (−3.76 − 16.3i)10-s + (2.52 + 7.76i)11-s + (11.1 − 5.66i)12-s + (8.08 + 15.8i)13-s + (−16.1 + 5.24i)14-s + (4.46 + 7.42i)15-s + (−2.20 + 6.77i)16-s + (4.14 + 26.1i)17-s + ⋯ |
| L(s) = 1 | + (0.759 − 1.49i)2-s + (−0.0903 + 0.570i)3-s + (−1.05 − 1.45i)4-s + (0.766 − 0.642i)5-s + (0.781 + 0.568i)6-s + (−0.512 − 0.512i)7-s + (−1.32 + 0.210i)8-s + (−0.317 − 0.103i)9-s + (−0.376 − 1.63i)10-s + (0.229 + 0.705i)11-s + (0.927 − 0.472i)12-s + (0.621 + 1.21i)13-s + (−1.15 + 0.374i)14-s + (0.297 + 0.494i)15-s + (−0.137 + 0.423i)16-s + (0.243 + 1.53i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.265 + 0.964i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.265 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.07379 - 1.40939i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.07379 - 1.40939i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.270 - 1.71i)T \) |
| 5 | \( 1 + (-3.83 + 3.21i)T \) |
| good | 2 | \( 1 + (-1.51 + 2.98i)T + (-2.35 - 3.23i)T^{2} \) |
| 7 | \( 1 + (3.58 + 3.58i)T + 49iT^{2} \) |
| 11 | \( 1 + (-2.52 - 7.76i)T + (-97.8 + 71.1i)T^{2} \) |
| 13 | \( 1 + (-8.08 - 15.8i)T + (-99.3 + 136. i)T^{2} \) |
| 17 | \( 1 + (-4.14 - 26.1i)T + (-274. + 89.3i)T^{2} \) |
| 19 | \( 1 + (-7.05 + 9.70i)T + (-111. - 343. i)T^{2} \) |
| 23 | \( 1 + (29.8 + 15.2i)T + (310. + 427. i)T^{2} \) |
| 29 | \( 1 + (-23.3 - 32.1i)T + (-259. + 799. i)T^{2} \) |
| 31 | \( 1 + (9.95 + 7.23i)T + (296. + 913. i)T^{2} \) |
| 37 | \( 1 + (29.4 - 15.0i)T + (804. - 1.10e3i)T^{2} \) |
| 41 | \( 1 + (-6.99 + 21.5i)T + (-1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 + (8.95 - 8.95i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (46.2 + 7.32i)T + (2.10e3 + 682. i)T^{2} \) |
| 53 | \( 1 + (10.1 - 64.2i)T + (-2.67e3 - 868. i)T^{2} \) |
| 59 | \( 1 + (-82.9 - 26.9i)T + (2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 + (24.5 + 75.4i)T + (-3.01e3 + 2.18e3i)T^{2} \) |
| 67 | \( 1 + (17.6 + 111. i)T + (-4.26e3 + 1.38e3i)T^{2} \) |
| 71 | \( 1 + (25.0 - 18.2i)T + (1.55e3 - 4.79e3i)T^{2} \) |
| 73 | \( 1 + (63.6 + 32.4i)T + (3.13e3 + 4.31e3i)T^{2} \) |
| 79 | \( 1 + (4.19 + 5.77i)T + (-1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (16.9 - 2.67i)T + (6.55e3 - 2.12e3i)T^{2} \) |
| 89 | \( 1 + (1.13 - 0.367i)T + (6.40e3 - 4.65e3i)T^{2} \) |
| 97 | \( 1 + (-82.1 - 13.0i)T + (8.94e3 + 2.90e3i)T^{2} \) |
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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
−13.73625073380603378359608121836, −12.76143287668345848546748929388, −11.93134582522841300977583816756, −10.58281914067333521554406302016, −9.936340658898646667966435684332, −8.880914217523281198737376232565, −6.30135907347032054651054532678, −4.75942773339302205165791742973, −3.75911501319999126938301860400, −1.74567634915111209173315263319,
3.19994612879728598402919719145, 5.55387060306157037433278687476, 6.08169074181986171026385951467, 7.25910154220293944123570484867, 8.411347755355931300911170024389, 9.941251348280978042985427247767, 11.67834636133643493497716661169, 13.02531775154166764171084240506, 13.75218665250341147106117118729, 14.43541122550890374286829605722
