| L(s) = 1 | + (−0.540 − 0.841i)2-s + (−1.59 + 0.671i)3-s + (−0.415 + 0.909i)4-s + (1.80 − 0.528i)5-s + (1.42 + 0.980i)6-s + (1.49 − 1.29i)7-s + (0.989 − 0.142i)8-s + (2.09 − 2.14i)9-s + (−1.41 − 1.22i)10-s + (1.85 + 1.19i)11-s + (0.0526 − 1.73i)12-s + (2.95 − 3.41i)13-s + (−1.89 − 0.555i)14-s + (−2.52 + 2.05i)15-s + (−0.654 − 0.755i)16-s + (0.301 + 0.660i)17-s + ⋯ |
| L(s) = 1 | + (−0.382 − 0.594i)2-s + (−0.921 + 0.387i)3-s + (−0.207 + 0.454i)4-s + (0.805 − 0.236i)5-s + (0.582 + 0.400i)6-s + (0.563 − 0.488i)7-s + (0.349 − 0.0503i)8-s + (0.699 − 0.714i)9-s + (−0.448 − 0.388i)10-s + (0.559 + 0.359i)11-s + (0.0151 − 0.499i)12-s + (0.819 − 0.945i)13-s + (−0.505 − 0.148i)14-s + (−0.650 + 0.530i)15-s + (−0.163 − 0.188i)16-s + (0.0731 + 0.160i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 + 0.646i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.762 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.786858 - 0.288510i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.786858 - 0.288510i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.540 + 0.841i)T \) |
| 3 | \( 1 + (1.59 - 0.671i)T \) |
| 23 | \( 1 + (3.69 - 3.05i)T \) |
| good | 5 | \( 1 + (-1.80 + 0.528i)T + (4.20 - 2.70i)T^{2} \) |
| 7 | \( 1 + (-1.49 + 1.29i)T + (0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (-1.85 - 1.19i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-2.95 + 3.41i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.301 - 0.660i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (0.501 + 0.229i)T + (12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-1.91 + 0.875i)T + (18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.655 - 4.55i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (-2.84 + 9.68i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-3.00 - 10.2i)T + (-34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (10.2 + 1.47i)T + (41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + 10.8iT - 47T^{2} \) |
| 53 | \( 1 + (1.98 + 2.28i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (4.87 + 4.22i)T + (8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (14.8 - 2.14i)T + (58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (-3.42 - 5.32i)T + (-27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (1.36 + 2.11i)T + (-29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (2.06 - 4.51i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (-10.6 - 9.20i)T + (11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (-10.5 - 3.10i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (0.391 - 2.72i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-1.95 - 6.66i)T + (-81.6 + 52.4i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.88957735174216920752475361687, −11.86739681217847424362197932247, −10.92370010506401079153122515705, −10.15490701395540211501421322788, −9.278907062099590686418748576685, −7.892629887707713122936640731696, −6.36320029341961523405891157760, −5.18019664741147028526640411024, −3.82962350982416990535100906542, −1.41584510186786201900774492333,
1.73568298850693279817892721808, 4.61277458711019723840815623037, 6.02908769213533410253585043727, 6.43971588639127746177849218715, 7.926883535612265163541312219428, 9.090159297357438323696376973281, 10.22994919627986822726264749112, 11.27296544065932102405273265909, 12.11197984977664639936873113382, 13.56219426929728504484297332594
