| L(s) = 1 | + (0.359 − 0.706i)2-s + (−0.270 + 1.71i)3-s + (1.98 + 2.72i)4-s + (−0.990 − 4.90i)5-s + (1.11 + 0.807i)6-s + (8.25 + 8.25i)7-s + (5.77 − 0.914i)8-s + (−2.85 − 0.927i)9-s + (−3.81 − 1.06i)10-s + (3.02 + 9.31i)11-s + (−5.20 + 2.65i)12-s + (−10.2 − 20.0i)13-s + (8.80 − 2.86i)14-s + (8.65 − 0.367i)15-s + (−2.73 + 8.42i)16-s + (−2.69 − 17.0i)17-s + ⋯ |
| L(s) = 1 | + (0.179 − 0.353i)2-s + (−0.0903 + 0.570i)3-s + (0.495 + 0.681i)4-s + (−0.198 − 0.980i)5-s + (0.185 + 0.134i)6-s + (1.17 + 1.17i)7-s + (0.721 − 0.114i)8-s + (−0.317 − 0.103i)9-s + (−0.381 − 0.106i)10-s + (0.275 + 0.847i)11-s + (−0.433 + 0.220i)12-s + (−0.784 − 1.54i)13-s + (0.629 − 0.204i)14-s + (0.576 − 0.0244i)15-s + (−0.171 + 0.526i)16-s + (−0.158 − 1.00i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 - 0.300i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.953 - 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.50261 + 0.230775i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.50261 + 0.230775i\) |
| \(L(2)\) |
|
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 | \( 1 + (0.990 + 4.90i)T \) |
| good | 2 | \( 1 + (-0.359 + 0.706i)T + (-2.35 - 3.23i)T^{2} \) |
| 7 | \( 1 + (-8.25 - 8.25i)T + 49iT^{2} \) |
| 11 | \( 1 + (-3.02 - 9.31i)T + (-97.8 + 71.1i)T^{2} \) |
| 13 | \( 1 + (10.2 + 20.0i)T + (-99.3 + 136. i)T^{2} \) |
| 17 | \( 1 + (2.69 + 17.0i)T + (-274. + 89.3i)T^{2} \) |
| 19 | \( 1 + (1.63 - 2.25i)T + (-111. - 343. i)T^{2} \) |
| 23 | \( 1 + (31.5 + 16.0i)T + (310. + 427. i)T^{2} \) |
| 29 | \( 1 + (-5.95 - 8.19i)T + (-259. + 799. i)T^{2} \) |
| 31 | \( 1 + (9.07 + 6.59i)T + (296. + 913. i)T^{2} \) |
| 37 | \( 1 + (-43.6 + 22.2i)T + (804. - 1.10e3i)T^{2} \) |
| 41 | \( 1 + (-1.68 + 5.18i)T + (-1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 + (-4.41 + 4.41i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (30.2 + 4.79i)T + (2.10e3 + 682. i)T^{2} \) |
| 53 | \( 1 + (-2.35 + 14.8i)T + (-2.67e3 - 868. i)T^{2} \) |
| 59 | \( 1 + (-74.9 - 24.3i)T + (2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 + (12.3 + 38.0i)T + (-3.01e3 + 2.18e3i)T^{2} \) |
| 67 | \( 1 + (-5.47 - 34.5i)T + (-4.26e3 + 1.38e3i)T^{2} \) |
| 71 | \( 1 + (77.2 - 56.1i)T + (1.55e3 - 4.79e3i)T^{2} \) |
| 73 | \( 1 + (-76.5 - 38.9i)T + (3.13e3 + 4.31e3i)T^{2} \) |
| 79 | \( 1 + (24.7 + 34.1i)T + (-1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (86.2 - 13.6i)T + (6.55e3 - 2.12e3i)T^{2} \) |
| 89 | \( 1 + (112. - 36.6i)T + (6.40e3 - 4.65e3i)T^{2} \) |
| 97 | \( 1 + (-50.7 - 8.03i)T + (8.94e3 + 2.90e3i)T^{2} \) |
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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
−14.55462923477783520244906653862, −12.79209109310070682079218910519, −12.13228658976371342991295517060, −11.38348537934996255916794248267, −9.908005466993683248425704766042, −8.535306739957969493716384057339, −7.68843402787914183578838787059, −5.42116650905610561188503905632, −4.40931040664721636146666988506, −2.37155754405772346677602953256,
1.78994398955660124362197649973, 4.28852789926644824685250455685, 6.12783108125850347095201933200, 7.07459026771948046854121937119, 7.982483537399857559665185969384, 10.06240854353079395714721340943, 11.16120858778465068534122530905, 11.62071581412446252787157940754, 13.74498658217983085253088396451, 14.21825672251084881907077751727
