| L(s) = 1 | + (−1.08 + 0.787i)2-s + (0.927 − 2.85i)3-s + (−1.91 + 5.90i)4-s + (−3.83 − 10.5i)5-s + (1.24 + 3.82i)6-s − 12.2·7-s + (−5.88 − 18.0i)8-s + (−7.28 − 5.29i)9-s + (12.4 + 8.36i)10-s + (2.00 − 1.45i)11-s + (15.0 + 10.9i)12-s + (−69.2 − 50.3i)13-s + (13.2 − 9.61i)14-s + (−33.5 + 1.19i)15-s + (−19.5 − 14.1i)16-s + (−1.71 − 5.27i)17-s + ⋯ |
| L(s) = 1 | + (−0.383 + 0.278i)2-s + (0.178 − 0.549i)3-s + (−0.239 + 0.737i)4-s + (−0.342 − 0.939i)5-s + (0.0845 + 0.260i)6-s − 0.659·7-s + (−0.259 − 0.799i)8-s + (−0.269 − 0.195i)9-s + (0.392 + 0.264i)10-s + (0.0550 − 0.0399i)11-s + (0.362 + 0.263i)12-s + (−1.47 − 1.07i)13-s + (0.252 − 0.183i)14-s + (−0.576 + 0.0206i)15-s + (−0.305 − 0.221i)16-s + (−0.0244 − 0.0752i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.678 + 0.734i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.678 + 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.193212 - 0.441298i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.193212 - 0.441298i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.927 + 2.85i)T \) |
| 5 | \( 1 + (3.83 + 10.5i)T \) |
| good | 2 | \( 1 + (1.08 - 0.787i)T + (2.47 - 7.60i)T^{2} \) |
| 7 | \( 1 + 12.2T + 343T^{2} \) |
| 11 | \( 1 + (-2.00 + 1.45i)T + (411. - 1.26e3i)T^{2} \) |
| 13 | \( 1 + (69.2 + 50.3i)T + (678. + 2.08e3i)T^{2} \) |
| 17 | \( 1 + (1.71 + 5.27i)T + (-3.97e3 + 2.88e3i)T^{2} \) |
| 19 | \( 1 + (7.45 + 22.9i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 + (-83.6 + 60.7i)T + (3.75e3 - 1.15e4i)T^{2} \) |
| 29 | \( 1 + (35.3 - 108. i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (-47.5 - 146. i)T + (-2.41e4 + 1.75e4i)T^{2} \) |
| 37 | \( 1 + (-277. - 201. i)T + (1.56e4 + 4.81e4i)T^{2} \) |
| 41 | \( 1 + (28.0 + 20.3i)T + (2.12e4 + 6.55e4i)T^{2} \) |
| 43 | \( 1 + 154.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-104. + 321. i)T + (-8.39e4 - 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-199. + 614. i)T + (-1.20e5 - 8.75e4i)T^{2} \) |
| 59 | \( 1 + (536. + 389. i)T + (6.34e4 + 1.95e5i)T^{2} \) |
| 61 | \( 1 + (-289. + 210. i)T + (7.01e4 - 2.15e5i)T^{2} \) |
| 67 | \( 1 + (164. + 505. i)T + (-2.43e5 + 1.76e5i)T^{2} \) |
| 71 | \( 1 + (196. - 603. i)T + (-2.89e5 - 2.10e5i)T^{2} \) |
| 73 | \( 1 + (779. - 565. i)T + (1.20e5 - 3.69e5i)T^{2} \) |
| 79 | \( 1 + (-175. + 540. i)T + (-3.98e5 - 2.89e5i)T^{2} \) |
| 83 | \( 1 + (98.8 + 304. i)T + (-4.62e5 + 3.36e5i)T^{2} \) |
| 89 | \( 1 + (-1.07e3 + 784. i)T + (2.17e5 - 6.70e5i)T^{2} \) |
| 97 | \( 1 + (-123. + 379. i)T + (-7.38e5 - 5.36e5i)T^{2} \) |
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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
−13.16181297920127755648590079413, −12.74702960089473816358979268323, −11.85425630224320709888046590625, −9.901456376470033100859765813955, −8.826148943800628397275154859558, −7.889119808927284447837477650447, −6.85622369258404367934790982353, −4.92788999550245332650459027205, −3.11441540155548608364361855070, −0.32851606819419246574153433126,
2.55787568898113425710359091875, 4.38003479015614342344747739740, 6.08104371947676307887277704717, 7.48561378267204131755997211267, 9.266712915363298048474243053550, 9.860960757191152509292537627425, 10.92148368129796059822084050550, 11.90881774612658171217182932801, 13.63222141423997588086090995489, 14.68750970783036459134554137220
