| L(s) = 1 | + (1.89 − 1.58i)2-s + (−1.45 + 0.942i)3-s + (0.711 − 4.03i)4-s + (2.16 − 0.788i)5-s + (−1.25 + 4.08i)6-s + (0.173 + 0.984i)7-s + (−2.59 − 4.48i)8-s + (1.22 − 2.73i)9-s + (2.84 − 4.93i)10-s + (−4.00 − 1.45i)11-s + (2.76 + 6.53i)12-s + (1.41 + 1.18i)13-s + (1.89 + 1.58i)14-s + (−2.40 + 3.18i)15-s + (−4.32 − 1.57i)16-s + (−2.58 + 4.47i)17-s + ⋯ |
| L(s) = 1 | + (1.33 − 1.12i)2-s + (−0.839 + 0.544i)3-s + (0.355 − 2.01i)4-s + (0.968 − 0.352i)5-s + (−0.511 + 1.66i)6-s + (0.0656 + 0.372i)7-s + (−0.916 − 1.58i)8-s + (0.407 − 0.913i)9-s + (0.900 − 1.55i)10-s + (−1.20 − 0.439i)11-s + (0.799 + 1.88i)12-s + (0.392 + 0.329i)13-s + (0.505 + 0.424i)14-s + (−0.620 + 0.823i)15-s + (−1.08 − 0.393i)16-s + (−0.626 + 1.08i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.211 + 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.211 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.52599 - 1.23076i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.52599 - 1.23076i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.45 - 0.942i)T \) |
| 7 | \( 1 + (-0.173 - 0.984i)T \) |
| good | 2 | \( 1 + (-1.89 + 1.58i)T + (0.347 - 1.96i)T^{2} \) |
| 5 | \( 1 + (-2.16 + 0.788i)T + (3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + (4.00 + 1.45i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-1.41 - 1.18i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (2.58 - 4.47i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.742 - 1.28i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.717 - 4.06i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-7.51 + 6.30i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (0.813 - 4.61i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (3.72 - 6.44i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (6.88 + 5.77i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (5.92 + 2.15i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (1.69 + 9.59i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 - 10.2T + 53T^{2} \) |
| 59 | \( 1 + (8.33 - 3.03i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.844 - 4.79i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-0.364 - 0.306i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-4.51 + 7.82i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (5.54 + 9.60i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.28 + 2.75i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-4.66 + 3.91i)T + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (5.05 + 8.76i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.34 - 1.21i)T + (74.3 + 62.3i)T^{2} \) |
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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
−12.23987240967692287712154373257, −11.60809514612948760471219217049, −10.41180499777114597496600096687, −10.16498660623010200655959038699, −8.692114110750666703820175269935, −6.32520740831552801173179803312, −5.56727921706633783129028145107, −4.83454175328636036548745981722, −3.47208859490100430762661419725, −1.82751247942138018953648542280,
2.64786502076228325653841775486, 4.68654580654164637354001501403, 5.39301895658166276265663164004, 6.44376273397509153558795001047, 7.08964673371214405805455249816, 8.146473394515260607277823383869, 10.06022996915000625956969313896, 11.02943230919845995498246811589, 12.28284953606816971687498800479, 13.11285520620257786697993870447
