| L(s) = 1 | + (1.40 − 0.179i)2-s + (0.537 + 1.64i)3-s + (1.93 − 0.502i)4-s + (−0.847 − 2.32i)5-s + (1.04 + 2.21i)6-s + (−4.59 + 0.810i)7-s + (2.62 − 1.05i)8-s + (−2.42 + 1.76i)9-s + (−1.60 − 3.11i)10-s + (2.23 + 0.812i)11-s + (1.86 + 2.91i)12-s + (−1.53 − 1.28i)13-s + (−6.30 + 1.95i)14-s + (3.38 − 2.64i)15-s + (3.49 − 1.94i)16-s + (1.59 + 0.920i)17-s + ⋯ |
| L(s) = 1 | + (0.991 − 0.126i)2-s + (0.310 + 0.950i)3-s + (0.967 − 0.251i)4-s + (−0.379 − 1.04i)5-s + (0.427 + 0.903i)6-s + (−1.73 + 0.306i)7-s + (0.928 − 0.371i)8-s + (−0.807 + 0.589i)9-s + (−0.508 − 0.985i)10-s + (0.673 + 0.244i)11-s + (0.538 + 0.842i)12-s + (−0.425 − 0.357i)13-s + (−1.68 + 0.523i)14-s + (0.872 − 0.683i)15-s + (0.873 − 0.486i)16-s + (0.386 + 0.223i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.209i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 - 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.62244 + 0.171709i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.62244 + 0.171709i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.40 + 0.179i)T \) |
| 3 | \( 1 + (-0.537 - 1.64i)T \) |
| good | 5 | \( 1 + (0.847 + 2.32i)T + (-3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (4.59 - 0.810i)T + (6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (-2.23 - 0.812i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (1.53 + 1.28i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.59 - 0.920i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.56 - 0.902i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.496 + 2.81i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-4.19 - 5.00i)T + (-5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (0.704 + 0.124i)T + (29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (5.60 - 9.71i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.52 + 5.39i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.206 + 0.568i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (0.587 + 3.32i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + 5.16iT - 53T^{2} \) |
| 59 | \( 1 + (7.69 - 2.79i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.01 - 5.75i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (0.427 - 0.509i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.933 + 1.61i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.519 - 0.899i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-8.75 - 10.4i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-2.22 + 1.86i)T + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (9.13 - 5.27i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (14.6 + 5.31i)T + (74.3 + 62.3i)T^{2} \) |
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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
−13.69196025558119862147423738489, −12.55903527892791406266185474524, −12.14368931057802001334373993714, −10.50838163773512428077928523916, −9.626802465540451237018905031005, −8.504841862974948530465995867315, −6.69146510533763207787383135641, −5.37831742784662212526853807724, −4.17855964214007208710893065055, −3.05238439594255604884081584402,
2.77891733434149887067214215792, 3.70348376572286534511707811553, 6.12611531947698700428691495325, 6.78008732998915253416978397413, 7.57291637088298710464765203712, 9.419475551557788823732278899078, 10.85305786221689515720417350840, 11.96098108973588623418724051476, 12.73941622269360760009586380792, 13.72665184175466748644312770743
