| L(s) = 1 | + (−1.27 − 1.54i)2-s + (−1.09 + 1.33i)3-s + (−0.755 + 3.92i)4-s + (0.143 + 0.472i)5-s + (3.46 − 0.0108i)6-s + (0.689 − 3.46i)7-s + (7.01 − 3.83i)8-s + (−0.585 − 2.94i)9-s + (0.545 − 0.822i)10-s + (−0.609 − 0.0600i)11-s + (−4.42 − 5.32i)12-s + (−6.44 − 1.95i)13-s + (−6.22 + 3.35i)14-s + (−0.789 − 0.327i)15-s + (−14.8 − 5.93i)16-s + (4.63 + 11.1i)17-s + ⋯ |
| L(s) = 1 | + (−0.636 − 0.771i)2-s + (−0.366 + 0.446i)3-s + (−0.188 + 0.981i)4-s + (0.0286 + 0.0944i)5-s + (0.577 − 0.00180i)6-s + (0.0985 − 0.495i)7-s + (0.877 − 0.479i)8-s + (−0.0650 − 0.326i)9-s + (0.0545 − 0.0822i)10-s + (−0.0553 − 0.00545i)11-s + (−0.369 − 0.443i)12-s + (−0.495 − 0.150i)13-s + (−0.444 + 0.239i)14-s + (−0.0526 − 0.0218i)15-s + (−0.928 − 0.371i)16-s + (0.272 + 0.658i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.997589 - 0.161858i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.997589 - 0.161858i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.27 + 1.54i)T \) |
| 3 | \( 1 + (1.09 - 1.33i)T \) |
| good | 5 | \( 1 + (-0.143 - 0.472i)T + (-20.7 + 13.8i)T^{2} \) |
| 7 | \( 1 + (-0.689 + 3.46i)T + (-45.2 - 18.7i)T^{2} \) |
| 11 | \( 1 + (0.609 + 0.0600i)T + (118. + 23.6i)T^{2} \) |
| 13 | \( 1 + (6.44 + 1.95i)T + (140. + 93.8i)T^{2} \) |
| 17 | \( 1 + (-4.63 - 11.1i)T + (-204. + 204. i)T^{2} \) |
| 19 | \( 1 + (6.61 - 3.53i)T + (200. - 300. i)T^{2} \) |
| 23 | \( 1 + (-17.2 - 11.4i)T + (202. + 488. i)T^{2} \) |
| 29 | \( 1 + (-34.4 + 3.39i)T + (824. - 164. i)T^{2} \) |
| 31 | \( 1 + (-12.7 + 12.7i)T - 961iT^{2} \) |
| 37 | \( 1 + (-7.06 - 3.77i)T + (760. + 1.13e3i)T^{2} \) |
| 41 | \( 1 + (-39.8 - 26.6i)T + (643. + 1.55e3i)T^{2} \) |
| 43 | \( 1 + (10.7 + 13.0i)T + (-360. + 1.81e3i)T^{2} \) |
| 47 | \( 1 + (-3.89 - 9.40i)T + (-1.56e3 + 1.56e3i)T^{2} \) |
| 53 | \( 1 + (-46.6 - 4.59i)T + (2.75e3 + 548. i)T^{2} \) |
| 59 | \( 1 + (-10.4 - 34.3i)T + (-2.89e3 + 1.93e3i)T^{2} \) |
| 61 | \( 1 + (-23.1 + 28.2i)T + (-725. - 3.64e3i)T^{2} \) |
| 67 | \( 1 + (39.8 + 32.7i)T + (875. + 4.40e3i)T^{2} \) |
| 71 | \( 1 + (-33.4 - 6.64i)T + (4.65e3 + 1.92e3i)T^{2} \) |
| 73 | \( 1 + (-76.0 + 15.1i)T + (4.92e3 - 2.03e3i)T^{2} \) |
| 79 | \( 1 + (-10.9 + 26.3i)T + (-4.41e3 - 4.41e3i)T^{2} \) |
| 83 | \( 1 + (14.9 + 27.9i)T + (-3.82e3 + 5.72e3i)T^{2} \) |
| 89 | \( 1 + (-65.4 - 97.8i)T + (-3.03e3 + 7.31e3i)T^{2} \) |
| 97 | \( 1 + (-7.49 - 7.49i)T + 9.40e3iT^{2} \) |
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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
−10.80454402051241011663130440577, −10.35153762266698166485244015175, −9.478789679277594888133925739450, −8.470501982811442994264884664875, −7.53745127634592736269748099943, −6.42461848347306949241897232710, −4.90079974813335043404518856399, −3.92601335399945866557741104641, −2.66288640164993779135075885421, −0.918026253421166051636831017452,
0.856466040644205610044067218929, 2.49214189680257563325817833640, 4.71247617807096470123982841455, 5.48921431334406262827944432767, 6.62626502515144438092514050180, 7.29072288025354525807215085579, 8.407871330040102506175393474323, 9.112275351089816772330989238996, 10.16709743769175188211076263551, 11.03859157558636629119830464702
