| L(s) = 1 | + (3.25 + 2.36i)2-s + (0.927 + 2.85i)3-s + (2.51 + 7.74i)4-s + (−6.00 + 9.42i)5-s + (−3.72 + 11.4i)6-s + 1.75·7-s + (−0.174 + 0.537i)8-s + (−7.28 + 5.29i)9-s + (−41.7 + 16.4i)10-s + (18.9 + 13.7i)11-s + (−19.7 + 14.3i)12-s + (43.1 − 31.3i)13-s + (5.71 + 4.15i)14-s + (−32.4 − 8.39i)15-s + (50.8 − 36.9i)16-s + (−6.45 + 19.8i)17-s + ⋯ |
| L(s) = 1 | + (1.14 + 0.834i)2-s + (0.178 + 0.549i)3-s + (0.314 + 0.967i)4-s + (−0.537 + 0.843i)5-s + (−0.253 + 0.779i)6-s + 0.0949·7-s + (−0.00771 + 0.0237i)8-s + (−0.269 + 0.195i)9-s + (−1.32 + 0.520i)10-s + (0.518 + 0.376i)11-s + (−0.475 + 0.345i)12-s + (0.919 − 0.668i)13-s + (0.109 + 0.0792i)14-s + (−0.558 − 0.144i)15-s + (0.794 − 0.577i)16-s + (−0.0921 + 0.283i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.305 - 0.952i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.52395 + 2.08987i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.52395 + 2.08987i\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + (6.00 - 9.42i)T \) |
| good | 2 | \( 1 + (-3.25 - 2.36i)T + (2.47 + 7.60i)T^{2} \) |
| 7 | \( 1 - 1.75T + 343T^{2} \) |
| 11 | \( 1 + (-18.9 - 13.7i)T + (411. + 1.26e3i)T^{2} \) |
| 13 | \( 1 + (-43.1 + 31.3i)T + (678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (6.45 - 19.8i)T + (-3.97e3 - 2.88e3i)T^{2} \) |
| 19 | \( 1 + (-5.67 + 17.4i)T + (-5.54e3 - 4.03e3i)T^{2} \) |
| 23 | \( 1 + (34.8 + 25.3i)T + (3.75e3 + 1.15e4i)T^{2} \) |
| 29 | \( 1 + (-41.2 - 126. i)T + (-1.97e4 + 1.43e4i)T^{2} \) |
| 31 | \( 1 + (-74.8 + 230. i)T + (-2.41e4 - 1.75e4i)T^{2} \) |
| 37 | \( 1 + (288. - 209. i)T + (1.56e4 - 4.81e4i)T^{2} \) |
| 41 | \( 1 + (-343. + 249. i)T + (2.12e4 - 6.55e4i)T^{2} \) |
| 43 | \( 1 + 93.5T + 7.95e4T^{2} \) |
| 47 | \( 1 + (72.1 + 222. i)T + (-8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (8.66 + 26.6i)T + (-1.20e5 + 8.75e4i)T^{2} \) |
| 59 | \( 1 + (365. - 265. i)T + (6.34e4 - 1.95e5i)T^{2} \) |
| 61 | \( 1 + (696. + 506. i)T + (7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 + (191. - 590. i)T + (-2.43e5 - 1.76e5i)T^{2} \) |
| 71 | \( 1 + (-292. - 898. i)T + (-2.89e5 + 2.10e5i)T^{2} \) |
| 73 | \( 1 + (-556. - 404. i)T + (1.20e5 + 3.69e5i)T^{2} \) |
| 79 | \( 1 + (173. + 532. i)T + (-3.98e5 + 2.89e5i)T^{2} \) |
| 83 | \( 1 + (290. - 895. i)T + (-4.62e5 - 3.36e5i)T^{2} \) |
| 89 | \( 1 + (912. + 663. i)T + (2.17e5 + 6.70e5i)T^{2} \) |
| 97 | \( 1 + (386. + 1.18e3i)T + (-7.38e5 + 5.36e5i)T^{2} \) |
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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
−14.49561012506004493443629516624, −13.69570804303654081117667855285, −12.43972436520543109921697823863, −11.16833556078191988578271665542, −10.02958708551579043587121322261, −8.294204580100044425681690515454, −7.02224195065259726319099805629, −5.90610826721121832969034984233, −4.38602746330873799749735665457, −3.32938049728235392536476223820,
1.48456118773690924978235268093, 3.45861772554793302415462561728, 4.66136577724005833737139766197, 6.16239025339332734227134327063, 7.970444613232591503750606674113, 9.085801796667503593124624059371, 10.99162357543214954917202790350, 11.85489738205245815252067053905, 12.55215622074787972137581663721, 13.62509841383911721085959946711
