| L(s) = 1 | + (2.36 − 1.84i)3-s + (−7.65 − 1.34i)5-s + (10.4 − 8.78i)7-s + (2.18 − 8.73i)9-s + (2.55 − 0.450i)11-s + (−8.23 + 2.99i)13-s + (−20.5 + 10.9i)15-s + (15.2 + 8.82i)17-s + (1.46 + 2.54i)19-s + (8.55 − 40.1i)21-s + (−11.8 + 14.1i)23-s + (33.2 + 12.1i)25-s + (−10.9 − 24.6i)27-s + (1.05 − 2.91i)29-s + (30.6 + 25.7i)31-s + ⋯ |
| L(s) = 1 | + (0.788 − 0.615i)3-s + (−1.53 − 0.269i)5-s + (1.49 − 1.25i)7-s + (0.242 − 0.970i)9-s + (0.232 − 0.0409i)11-s + (−0.633 + 0.230i)13-s + (−1.37 + 0.728i)15-s + (0.899 + 0.519i)17-s + (0.0773 + 0.133i)19-s + (0.407 − 1.91i)21-s + (−0.517 + 0.616i)23-s + (1.33 + 0.484i)25-s + (−0.405 − 0.914i)27-s + (0.0365 − 0.100i)29-s + (0.988 + 0.829i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.296 + 0.955i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.296 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.22115 - 0.899609i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.22115 - 0.899609i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.36 + 1.84i)T \) |
| good | 5 | \( 1 + (7.65 + 1.34i)T + (23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (-10.4 + 8.78i)T + (8.50 - 48.2i)T^{2} \) |
| 11 | \( 1 + (-2.55 + 0.450i)T + (113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (8.23 - 2.99i)T + (129. - 108. i)T^{2} \) |
| 17 | \( 1 + (-15.2 - 8.82i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-1.46 - 2.54i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (11.8 - 14.1i)T + (-91.8 - 520. i)T^{2} \) |
| 29 | \( 1 + (-1.05 + 2.91i)T + (-644. - 540. i)T^{2} \) |
| 31 | \( 1 + (-30.6 - 25.7i)T + (166. + 946. i)T^{2} \) |
| 37 | \( 1 + (12.8 - 22.2i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-21.2 - 58.3i)T + (-1.28e3 + 1.08e3i)T^{2} \) |
| 43 | \( 1 + (-5.71 - 32.4i)T + (-1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (33.4 + 39.8i)T + (-383. + 2.17e3i)T^{2} \) |
| 53 | \( 1 + 53.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-102. - 18.0i)T + (3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (4.56 - 3.82i)T + (646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (44.4 - 16.1i)T + (3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (77.5 + 44.7i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-33.6 - 58.2i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-34.1 - 12.4i)T + (4.78e3 + 4.01e3i)T^{2} \) |
| 83 | \( 1 + (-6.94 + 19.0i)T + (-5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (-83.7 + 48.3i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (11.6 + 66.2i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
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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
−13.35380166623841321889150722040, −12.03008718739786579682735859351, −11.50800899524496988534217284245, −10.07776170155724553975698461138, −8.307607145138798503306511969963, −7.912450748209270560786723771170, −7.03823698968335976728029820317, −4.63478479727764350305382129927, −3.63230856485799713348279019666, −1.23150171543428480990526725699,
2.57832448782964776518950783478, 4.13860763918609623658883782987, 5.22518868344146846839557025412, 7.57754002580059686748993246304, 8.136947596550386865799256412482, 9.129598024006038321038493055714, 10.62125637923530805525962594022, 11.68557822970619108973793568989, 12.25629345037647391852593610250, 14.17096920371993114907823340493
