| L(s) = 1 | + (1.73 − 2.06i)2-s + (−1.71 − 0.239i)3-s + (−0.918 − 5.20i)4-s + (1.79 + 1.33i)5-s + (−3.47 + 3.13i)6-s + (−0.996 − 0.175i)7-s + (−7.68 − 4.43i)8-s + (2.88 + 0.820i)9-s + (5.86 − 1.40i)10-s + (−1.18 + 0.430i)11-s + (0.329 + 9.15i)12-s + (2.66 + 3.17i)13-s + (−2.09 + 1.75i)14-s + (−2.76 − 2.71i)15-s + (−12.5 + 4.57i)16-s + (3.38 − 1.95i)17-s + ⋯ |
| L(s) = 1 | + (1.22 − 1.46i)2-s + (−0.990 − 0.138i)3-s + (−0.459 − 2.60i)4-s + (0.803 + 0.594i)5-s + (−1.41 + 1.27i)6-s + (−0.376 − 0.0664i)7-s + (−2.71 − 1.56i)8-s + (0.961 + 0.273i)9-s + (1.85 − 0.445i)10-s + (−0.356 + 0.129i)11-s + (0.0952 + 2.64i)12-s + (0.739 + 0.880i)13-s + (−0.559 + 0.469i)14-s + (−0.714 − 0.700i)15-s + (−3.14 + 1.14i)16-s + (0.821 − 0.474i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.727548 - 1.35804i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.727548 - 1.35804i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.71 + 0.239i)T \) |
| 5 | \( 1 + (-1.79 - 1.33i)T \) |
| good | 2 | \( 1 + (-1.73 + 2.06i)T + (-0.347 - 1.96i)T^{2} \) |
| 7 | \( 1 + (0.996 + 0.175i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (1.18 - 0.430i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-2.66 - 3.17i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-3.38 + 1.95i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.433 - 0.750i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.71 + 0.479i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (6.00 + 5.03i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.941 - 5.33i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (6.44 - 3.71i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.73 - 2.29i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-1.45 - 4.00i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (3.63 + 0.640i)T + (44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + 0.485iT - 53T^{2} \) |
| 59 | \( 1 + (1.47 + 0.538i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.286 + 1.62i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-2.13 - 2.53i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (4.95 + 8.58i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.55 - 2.63i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.41 - 5.38i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-1.86 + 2.21i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-8.03 + 13.9i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.05 + 2.90i)T + (-74.3 + 62.3i)T^{2} \) |
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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
−12.86999364950997032783031187182, −11.82233587567813082035921200389, −11.08608389121752917989487944441, −10.23947365779367933296630441468, −9.515726981143564130837656560595, −6.76201707576607669055786673813, −5.87104410107050091842046393024, −4.86200436155790884034278770109, −3.35233408408533287495324376815, −1.69354024705677503042471767599,
3.64217206045537282052544595562, 5.16935239396017928153847130586, 5.67280275858741079878508541890, 6.59965813153197879754673124123, 7.87175115384930466709756609211, 9.164633620565423006128896229129, 10.65080065497915811962153770248, 12.14466173553907288833536816372, 12.94750405758173075612339999897, 13.36450558545405691898207286068
