| L(s) = 1 | + (−3.76 + 2.73i)2-s + (0.927 − 2.85i)3-s + (4.22 − 12.9i)4-s + (5.34 + 9.82i)5-s + (4.31 + 13.2i)6-s − 26.0·7-s + (8.14 + 25.0i)8-s + (−7.28 − 5.29i)9-s + (−46.9 − 22.3i)10-s + (−2.03 + 1.47i)11-s + (−33.1 − 24.0i)12-s + (−32.0 − 23.2i)13-s + (98.0 − 71.2i)14-s + (32.9 − 6.13i)15-s + (−10.7 − 7.84i)16-s + (−26.2 − 80.8i)17-s + ⋯ |
| L(s) = 1 | + (−1.33 + 0.967i)2-s + (0.178 − 0.549i)3-s + (0.527 − 1.62i)4-s + (0.477 + 0.878i)5-s + (0.293 + 0.903i)6-s − 1.40·7-s + (0.359 + 1.10i)8-s + (−0.269 − 0.195i)9-s + (−1.48 − 0.707i)10-s + (−0.0557 + 0.0405i)11-s + (−0.797 − 0.579i)12-s + (−0.683 − 0.496i)13-s + (1.87 − 1.36i)14-s + (0.567 − 0.105i)15-s + (−0.168 − 0.122i)16-s + (−0.374 − 1.15i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.434 + 0.900i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.434 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0614836 - 0.0978781i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0614836 - 0.0978781i\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + (-5.34 - 9.82i)T \) |
| good | 2 | \( 1 + (3.76 - 2.73i)T + (2.47 - 7.60i)T^{2} \) |
| 7 | \( 1 + 26.0T + 343T^{2} \) |
| 11 | \( 1 + (2.03 - 1.47i)T + (411. - 1.26e3i)T^{2} \) |
| 13 | \( 1 + (32.0 + 23.2i)T + (678. + 2.08e3i)T^{2} \) |
| 17 | \( 1 + (26.2 + 80.8i)T + (-3.97e3 + 2.88e3i)T^{2} \) |
| 19 | \( 1 + (44.1 + 135. i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 + (127. - 92.3i)T + (3.75e3 - 1.15e4i)T^{2} \) |
| 29 | \( 1 + (30.9 - 95.3i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (53.5 + 164. i)T + (-2.41e4 + 1.75e4i)T^{2} \) |
| 37 | \( 1 + (-48.0 - 34.9i)T + (1.56e4 + 4.81e4i)T^{2} \) |
| 41 | \( 1 + (-287. - 208. i)T + (2.12e4 + 6.55e4i)T^{2} \) |
| 43 | \( 1 - 109.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (17.9 - 55.2i)T + (-8.39e4 - 6.10e4i)T^{2} \) |
| 53 | \( 1 + (120. - 371. i)T + (-1.20e5 - 8.75e4i)T^{2} \) |
| 59 | \( 1 + (333. + 242. i)T + (6.34e4 + 1.95e5i)T^{2} \) |
| 61 | \( 1 + (290. - 210. i)T + (7.01e4 - 2.15e5i)T^{2} \) |
| 67 | \( 1 + (-108. - 333. i)T + (-2.43e5 + 1.76e5i)T^{2} \) |
| 71 | \( 1 + (-52.7 + 162. i)T + (-2.89e5 - 2.10e5i)T^{2} \) |
| 73 | \( 1 + (-754. + 548. i)T + (1.20e5 - 3.69e5i)T^{2} \) |
| 79 | \( 1 + (405. - 1.24e3i)T + (-3.98e5 - 2.89e5i)T^{2} \) |
| 83 | \( 1 + (202. + 621. i)T + (-4.62e5 + 3.36e5i)T^{2} \) |
| 89 | \( 1 + (857. - 622. i)T + (2.17e5 - 6.70e5i)T^{2} \) |
| 97 | \( 1 + (-198. + 610. i)T + (-7.38e5 - 5.36e5i)T^{2} \) |
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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
−13.86639982987302077987530298917, −12.82435648384431977771505464033, −11.06881576744969420288924423199, −9.748425784824275111937189037557, −9.282107738450133027158236395671, −7.58516307003950720572588694057, −6.86504669344016062607097755182, −5.94798728433334089882826643050, −2.71389378085298681191865116487, −0.10226905548840861599519925091,
2.10123461946117215141554996427, 3.89692856782256680636782397563, 6.09372038626848118563720842922, 8.107803542271979988922937011106, 9.090329664270663703973076921761, 9.891310426553066424784667799507, 10.51710818445431264007776665258, 12.22641993805927075246399627074, 12.74147983768745450502786157353, 14.31839391374782535015949736495
