| L(s) = 1 | + (0.326 + 1.37i)2-s + (−2.78 + 1.11i)3-s + (−1.78 + 0.897i)4-s + (5.92 − 4.41i)5-s + (−2.44 − 3.46i)6-s + (6.83 − 4.49i)7-s + (−1.81 − 2.16i)8-s + (6.50 − 6.21i)9-s + (8.00 + 6.71i)10-s + (−1.35 + 11.6i)11-s + (3.97 − 4.49i)12-s + (0.463 + 1.54i)13-s + (8.41 + 7.93i)14-s + (−11.5 + 18.8i)15-s + (2.38 − 3.20i)16-s + (12.5 + 2.21i)17-s + ⋯ |
| L(s) = 1 | + (0.163 + 0.688i)2-s + (−0.928 + 0.372i)3-s + (−0.446 + 0.224i)4-s + (1.18 − 0.882i)5-s + (−0.407 − 0.577i)6-s + (0.976 − 0.642i)7-s + (−0.227 − 0.270i)8-s + (0.722 − 0.691i)9-s + (0.800 + 0.671i)10-s + (−0.123 + 1.05i)11-s + (0.331 − 0.374i)12-s + (0.0356 + 0.119i)13-s + (0.601 + 0.567i)14-s + (−0.771 + 1.25i)15-s + (0.149 − 0.200i)16-s + (0.739 + 0.130i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.782 - 0.623i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.782 - 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.43779 + 0.502651i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.43779 + 0.502651i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.326 - 1.37i)T \) |
| 3 | \( 1 + (2.78 - 1.11i)T \) |
| good | 5 | \( 1 + (-5.92 + 4.41i)T + (7.17 - 23.9i)T^{2} \) |
| 7 | \( 1 + (-6.83 + 4.49i)T + (19.4 - 44.9i)T^{2} \) |
| 11 | \( 1 + (1.35 - 11.6i)T + (-117. - 27.9i)T^{2} \) |
| 13 | \( 1 + (-0.463 - 1.54i)T + (-141. + 92.8i)T^{2} \) |
| 17 | \( 1 + (-12.5 - 2.21i)T + (271. + 98.8i)T^{2} \) |
| 19 | \( 1 + (-2.62 - 14.9i)T + (-339. + 123. i)T^{2} \) |
| 23 | \( 1 + (-14.0 + 21.4i)T + (-209. - 485. i)T^{2} \) |
| 29 | \( 1 + (-35.0 + 33.1i)T + (48.8 - 839. i)T^{2} \) |
| 31 | \( 1 + (0.496 - 8.53i)T + (-954. - 111. i)T^{2} \) |
| 37 | \( 1 + (59.1 - 21.5i)T + (1.04e3 - 879. i)T^{2} \) |
| 41 | \( 1 + (-10.1 + 42.6i)T + (-1.50e3 - 754. i)T^{2} \) |
| 43 | \( 1 + (-9.46 - 21.9i)T + (-1.26e3 + 1.34e3i)T^{2} \) |
| 47 | \( 1 + (39.0 - 2.27i)T + (2.19e3 - 256. i)T^{2} \) |
| 53 | \( 1 + (-57.7 - 33.3i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-3.88 - 33.2i)T + (-3.38e3 + 802. i)T^{2} \) |
| 61 | \( 1 + (87.3 + 43.8i)T + (2.22e3 + 2.98e3i)T^{2} \) |
| 67 | \( 1 + (-23.9 + 25.3i)T + (-261. - 4.48e3i)T^{2} \) |
| 71 | \( 1 + (57.1 - 68.1i)T + (-875. - 4.96e3i)T^{2} \) |
| 73 | \( 1 + (-24.8 + 20.8i)T + (925. - 5.24e3i)T^{2} \) |
| 79 | \( 1 + (116. - 27.5i)T + (5.57e3 - 2.80e3i)T^{2} \) |
| 83 | \( 1 + (32.1 + 135. i)T + (-6.15e3 + 3.09e3i)T^{2} \) |
| 89 | \( 1 + (112. + 134. i)T + (-1.37e3 + 7.80e3i)T^{2} \) |
| 97 | \( 1 + (57.6 - 77.4i)T + (-2.69e3 - 9.01e3i)T^{2} \) |
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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
−12.69530890241558767319268874478, −11.99905536046905293948195854348, −10.44227227326919154338581242513, −9.860778850208113744345586322117, −8.618874609748716148650338410779, −7.30797898285645159362362682550, −6.08272527936585092899415921399, −5.07418795092594309392386253437, −4.41600746163561627662068927720, −1.36202054642317374953938595201,
1.47683290973553796517965266213, 2.90662938834251053570404409191, 5.14822713798560783976988839736, 5.74245564738768180223165914277, 6.99319606858925989717603051482, 8.536371680140217558479980459359, 9.860216585110013728259280466368, 10.82954601878588787787884842389, 11.34433128588592954062449464703, 12.35628477019149565625445831310
