| L(s) = 1 | + (0.326 − 1.37i)2-s + (2.18 − 2.05i)3-s + (−1.78 − 0.897i)4-s + (−1.03 − 0.772i)5-s + (−2.10 − 3.68i)6-s + (−4.52 − 2.97i)7-s + (−1.81 + 2.16i)8-s + (0.588 − 8.98i)9-s + (−1.40 + 1.17i)10-s + (−0.852 − 7.29i)11-s + (−5.75 + 1.69i)12-s + (0.717 − 2.39i)13-s + (−5.57 + 5.26i)14-s + (−3.85 + 0.436i)15-s + (2.38 + 3.20i)16-s + (12.8 − 2.26i)17-s + ⋯ |
| L(s) = 1 | + (0.163 − 0.688i)2-s + (0.729 − 0.683i)3-s + (−0.446 − 0.224i)4-s + (−0.207 − 0.154i)5-s + (−0.351 − 0.613i)6-s + (−0.647 − 0.425i)7-s + (−0.227 + 0.270i)8-s + (0.0654 − 0.997i)9-s + (−0.140 + 0.117i)10-s + (−0.0774 − 0.662i)11-s + (−0.479 + 0.141i)12-s + (0.0551 − 0.184i)13-s + (−0.398 + 0.375i)14-s + (−0.257 + 0.0290i)15-s + (0.149 + 0.200i)16-s + (0.754 − 0.132i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.750 + 0.661i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.750 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.579342 - 1.53364i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.579342 - 1.53364i\) |
| \(L(2)\) |
|
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 | \( 1 + (-2.18 + 2.05i)T \) |
| good | 5 | \( 1 + (1.03 + 0.772i)T + (7.17 + 23.9i)T^{2} \) |
| 7 | \( 1 + (4.52 + 2.97i)T + (19.4 + 44.9i)T^{2} \) |
| 11 | \( 1 + (0.852 + 7.29i)T + (-117. + 27.9i)T^{2} \) |
| 13 | \( 1 + (-0.717 + 2.39i)T + (-141. - 92.8i)T^{2} \) |
| 17 | \( 1 + (-12.8 + 2.26i)T + (271. - 98.8i)T^{2} \) |
| 19 | \( 1 + (0.163 - 0.928i)T + (-339. - 123. i)T^{2} \) |
| 23 | \( 1 + (-23.9 - 36.3i)T + (-209. + 485. i)T^{2} \) |
| 29 | \( 1 + (-10.5 - 9.97i)T + (48.8 + 839. i)T^{2} \) |
| 31 | \( 1 + (0.520 + 8.93i)T + (-954. + 111. i)T^{2} \) |
| 37 | \( 1 + (-18.2 - 6.63i)T + (1.04e3 + 879. i)T^{2} \) |
| 41 | \( 1 + (-2.27 - 9.59i)T + (-1.50e3 + 754. i)T^{2} \) |
| 43 | \( 1 + (-2.47 + 5.74i)T + (-1.26e3 - 1.34e3i)T^{2} \) |
| 47 | \( 1 + (-64.0 - 3.72i)T + (2.19e3 + 256. i)T^{2} \) |
| 53 | \( 1 + (-31.0 + 17.9i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (12.6 - 108. i)T + (-3.38e3 - 802. i)T^{2} \) |
| 61 | \( 1 + (-55.2 + 27.7i)T + (2.22e3 - 2.98e3i)T^{2} \) |
| 67 | \( 1 + (-8.81 - 9.34i)T + (-261. + 4.48e3i)T^{2} \) |
| 71 | \( 1 + (-9.13 - 10.8i)T + (-875. + 4.96e3i)T^{2} \) |
| 73 | \( 1 + (-31.7 - 26.6i)T + (925. + 5.24e3i)T^{2} \) |
| 79 | \( 1 + (76.6 + 18.1i)T + (5.57e3 + 2.80e3i)T^{2} \) |
| 83 | \( 1 + (0.889 - 3.75i)T + (-6.15e3 - 3.09e3i)T^{2} \) |
| 89 | \( 1 + (-7.81 + 9.31i)T + (-1.37e3 - 7.80e3i)T^{2} \) |
| 97 | \( 1 + (25.5 + 34.2i)T + (-2.69e3 + 9.01e3i)T^{2} \) |
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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
−12.36296033579070932868275694628, −11.45455770392296862901168594193, −10.17173568189999572413222686505, −9.232840693063506326557208986957, −8.179593776609689806941487350039, −7.08778850932232550180586294353, −5.70235756250974964381422170061, −3.85284000285783488235575952215, −2.86779025421993224346103860664, −0.965763838215728270979713613647,
2.79151174323904941281768777820, 4.10354830552362013777558628335, 5.32142272317336864980057912150, 6.75552621507181357226019924408, 7.85727171908937991134329094324, 8.929878261083549850761755479780, 9.719056128518928116633357227912, 10.78658050174289995532716025560, 12.30579385795193900956946736143, 13.14448311693762407212347997521
