L(s) = 1 | + (0.622 − 2.75i)2-s + (−1.95 − 4.81i)3-s + (−7.22 − 3.43i)4-s + 5i·5-s + (−14.4 + 2.39i)6-s − 20.3i·7-s + (−13.9 + 17.7i)8-s + (−19.3 + 18.8i)9-s + (13.7 + 3.11i)10-s − 15.2·11-s + (−2.40 + 41.4i)12-s + 27.7·13-s + (−56.1 − 12.6i)14-s + (24.0 − 9.77i)15-s + (40.4 + 49.6i)16-s − 92.5i·17-s + ⋯ |
L(s) = 1 | + (0.220 − 0.975i)2-s + (−0.376 − 0.926i)3-s + (−0.903 − 0.429i)4-s + 0.447i·5-s + (−0.986 + 0.163i)6-s − 1.09i·7-s + (−0.617 + 0.786i)8-s + (−0.716 + 0.697i)9-s + (0.436 + 0.0984i)10-s − 0.419·11-s + (−0.0578 + 0.998i)12-s + 0.591·13-s + (−1.07 − 0.241i)14-s + (0.414 − 0.168i)15-s + (0.631 + 0.775i)16-s − 1.31i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0578i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0578i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.0299325 + 1.03327i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0299325 + 1.03327i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.622 + 2.75i)T \) |
| 3 | \( 1 + (1.95 + 4.81i)T \) |
| 5 | \( 1 - 5iT \) |
good | 7 | \( 1 + 20.3iT - 343T^{2} \) |
| 11 | \( 1 + 15.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 27.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 92.5iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 127. iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 51.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 99.2iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 25.8iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 356.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 292. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 521. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 573.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 305. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 295.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 326.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 299. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 653.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 504.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 110. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 1.47e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 772. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 26.1T + 9.12e5T^{2} \) |
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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
−13.42572580316877394882399989106, −13.20311272622801987712112840264, −11.46029141292181729609271421339, −11.05702984610228298717049125605, −9.634726215538468689231461087688, −7.902623452485460294309359066291, −6.56708573748516299997210841478, −4.81098877791758578433119661944, −2.81245786014547006769249703361, −0.74152264348654388130164651475,
3.78420347620215235584754056365, 5.31833068817796178468502125852, 6.10893995033056460614645147042, 8.235258875179545312063239319917, 9.044066893816807119538915076977, 10.36314962857268960401427007427, 12.00376326890504176999713597635, 12.93246328502256198975503645523, 14.50726943306030497453739859400, 15.27329860590427318925572536748