| L(s) = 1 | + (0.173 − 0.984i)2-s + (2.00 + 1.68i)3-s + (−0.939 − 0.342i)4-s + (0.939 − 0.342i)5-s + (2.00 − 1.68i)6-s + (−0.485 − 0.840i)7-s + (−0.5 + 0.866i)8-s + (0.668 + 3.79i)9-s + (−0.173 − 0.984i)10-s + (0.280 − 0.486i)11-s + (−1.30 − 2.26i)12-s + (−0.350 + 0.293i)13-s + (−0.911 + 0.331i)14-s + (2.45 + 0.895i)15-s + (0.766 + 0.642i)16-s + (0.0682 − 0.387i)17-s + ⋯ |
| L(s) = 1 | + (0.122 − 0.696i)2-s + (1.15 + 0.971i)3-s + (−0.469 − 0.171i)4-s + (0.420 − 0.152i)5-s + (0.818 − 0.686i)6-s + (−0.183 − 0.317i)7-s + (−0.176 + 0.306i)8-s + (0.222 + 1.26i)9-s + (−0.0549 − 0.311i)10-s + (0.0846 − 0.146i)11-s + (−0.377 − 0.654i)12-s + (−0.0971 + 0.0815i)13-s + (−0.243 + 0.0887i)14-s + (0.635 + 0.231i)15-s + (0.191 + 0.160i)16-s + (0.0165 − 0.0938i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.212i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 + 0.212i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.67693 - 0.180561i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.67693 - 0.180561i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.173 + 0.984i)T \) |
| 5 | \( 1 + (-0.939 + 0.342i)T \) |
| 19 | \( 1 + (1.58 - 4.05i)T \) |
| good | 3 | \( 1 + (-2.00 - 1.68i)T + (0.520 + 2.95i)T^{2} \) |
| 7 | \( 1 + (0.485 + 0.840i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.280 + 0.486i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.350 - 0.293i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.0682 + 0.387i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (4.92 + 1.79i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (0.411 + 2.33i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (5.44 + 9.42i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 5.14T + 37T^{2} \) |
| 41 | \( 1 + (-6.14 - 5.15i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (9.61 - 3.49i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-2.13 - 12.0i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-4.26 - 1.55i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (-0.759 + 4.30i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-4.02 - 1.46i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-2.11 - 12.0i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-6.85 + 2.49i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (10.6 + 8.96i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (0.730 + 0.612i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-5.63 - 9.76i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-2.69 + 2.26i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-1.25 + 7.10i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−12.67231728053165919849877752842, −11.40106676649262998176589285681, −10.20243755748376564183990262223, −9.757750685661223082004760430727, −8.804353628124523059241921991954, −7.84339015614680560475660605848, −5.96865667888082866202679765610, −4.44154087174729065495763730590, −3.58353461824874530635815244859, −2.21987047094995970249900420211,
2.07112712898942600092008390190, 3.49086525675509767427650807320, 5.33377900575864441707857410135, 6.68643733264572068422289793877, 7.34860371487743234152954612015, 8.548991959853359151077725697409, 9.100858301385589462022326250975, 10.39876950223743443648464142217, 12.07667354666222181894387258086, 12.89904083048986603227554507051
