| L(s) = 1 | + (−0.326 − 1.37i)2-s + (−0.663 + 2.92i)3-s + (−1.78 + 0.897i)4-s + (0.605 − 0.450i)5-s + (4.24 − 0.0416i)6-s + (1.59 − 1.05i)7-s + (1.81 + 2.16i)8-s + (−8.12 − 3.88i)9-s + (−0.817 − 0.686i)10-s + (−1.86 + 15.9i)11-s + (−1.44 − 5.82i)12-s + (5.81 + 19.4i)13-s + (−1.96 − 1.85i)14-s + (0.917 + 2.07i)15-s + (2.38 − 3.20i)16-s + (−24.4 − 4.30i)17-s + ⋯ |
| L(s) = 1 | + (−0.163 − 0.688i)2-s + (−0.221 + 0.975i)3-s + (−0.446 + 0.224i)4-s + (0.121 − 0.0901i)5-s + (0.707 − 0.00694i)6-s + (0.228 − 0.150i)7-s + (0.227 + 0.270i)8-s + (−0.902 − 0.431i)9-s + (−0.0817 − 0.0686i)10-s + (−0.169 + 1.45i)11-s + (−0.120 − 0.485i)12-s + (0.447 + 1.49i)13-s + (−0.140 − 0.132i)14-s + (0.0611 + 0.138i)15-s + (0.149 − 0.200i)16-s + (−1.43 − 0.253i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.753958 + 0.632972i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.753958 + 0.632972i\) |
| \(L(2)\) |
|
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 | \( 1 + (0.663 - 2.92i)T \) |
| good | 5 | \( 1 + (-0.605 + 0.450i)T + (7.17 - 23.9i)T^{2} \) |
| 7 | \( 1 + (-1.59 + 1.05i)T + (19.4 - 44.9i)T^{2} \) |
| 11 | \( 1 + (1.86 - 15.9i)T + (-117. - 27.9i)T^{2} \) |
| 13 | \( 1 + (-5.81 - 19.4i)T + (-141. + 92.8i)T^{2} \) |
| 17 | \( 1 + (24.4 + 4.30i)T + (271. + 98.8i)T^{2} \) |
| 19 | \( 1 + (-5.59 - 31.7i)T + (-339. + 123. i)T^{2} \) |
| 23 | \( 1 + (-4.73 + 7.19i)T + (-209. - 485. i)T^{2} \) |
| 29 | \( 1 + (-14.0 + 13.2i)T + (48.8 - 839. i)T^{2} \) |
| 31 | \( 1 + (-3.26 + 56.0i)T + (-954. - 111. i)T^{2} \) |
| 37 | \( 1 + (-22.6 + 8.23i)T + (1.04e3 - 879. i)T^{2} \) |
| 41 | \( 1 + (-2.75 + 11.6i)T + (-1.50e3 - 754. i)T^{2} \) |
| 43 | \( 1 + (-24.1 - 56.0i)T + (-1.26e3 + 1.34e3i)T^{2} \) |
| 47 | \( 1 + (-21.9 + 1.27i)T + (2.19e3 - 256. i)T^{2} \) |
| 53 | \( 1 + (68.0 + 39.2i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (2.00 + 17.1i)T + (-3.38e3 + 802. i)T^{2} \) |
| 61 | \( 1 + (0.791 + 0.397i)T + (2.22e3 + 2.98e3i)T^{2} \) |
| 67 | \( 1 + (-49.4 + 52.3i)T + (-261. - 4.48e3i)T^{2} \) |
| 71 | \( 1 + (-57.3 + 68.3i)T + (-875. - 4.96e3i)T^{2} \) |
| 73 | \( 1 + (-24.8 + 20.8i)T + (925. - 5.24e3i)T^{2} \) |
| 79 | \( 1 + (10.8 - 2.57i)T + (5.57e3 - 2.80e3i)T^{2} \) |
| 83 | \( 1 + (-16.3 - 68.8i)T + (-6.15e3 + 3.09e3i)T^{2} \) |
| 89 | \( 1 + (2.69 + 3.21i)T + (-1.37e3 + 7.80e3i)T^{2} \) |
| 97 | \( 1 + (8.09 - 10.8i)T + (-2.69e3 - 9.01e3i)T^{2} \) |
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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
−12.62630945488503812027198710016, −11.57620631098481894311984536055, −10.94378919038716157615120536438, −9.670344186049594134971772732050, −9.338038355262916238424292529312, −7.895141620027788890872961595386, −6.30619213597530751955441276089, −4.69569442610962135090022052302, −4.01337094970869281943086864137, −2.06591506905264366623502102975,
0.67383361823875968678342782153, 2.89347000185202918385657433843, 5.11879722947257854413708712586, 6.10157942280518116349062025807, 7.02723784164130185815260991944, 8.318852148387408208398998811890, 8.764890535788362958508053365859, 10.63849693922429578583774722858, 11.28659053639014697725718976737, 12.70738268023129276354786182188
