| L(s) = 1 | + (0.888 + 0.458i)2-s + (−1.91 + 0.563i)3-s + (0.580 + 0.814i)4-s + (1.21 + 1.40i)5-s + (−1.96 − 0.378i)6-s + (0.0846 + 1.77i)7-s + (0.142 + 0.989i)8-s + (0.838 − 0.538i)9-s + (0.438 + 1.80i)10-s + (−0.216 + 0.0417i)11-s + (−1.57 − 1.23i)12-s + (1.83 − 0.735i)13-s + (−0.739 + 1.61i)14-s + (−3.12 − 2.00i)15-s + (−0.327 + 0.945i)16-s + (3.10 − 4.36i)17-s + ⋯ |
| L(s) = 1 | + (0.628 + 0.324i)2-s + (−1.10 + 0.325i)3-s + (0.290 + 0.407i)4-s + (0.544 + 0.628i)5-s + (−0.801 − 0.154i)6-s + (0.0319 + 0.671i)7-s + (0.0503 + 0.349i)8-s + (0.279 − 0.179i)9-s + (0.138 + 0.571i)10-s + (−0.0652 + 0.0125i)11-s + (−0.453 − 0.356i)12-s + (0.509 − 0.203i)13-s + (−0.197 + 0.432i)14-s + (−0.806 − 0.518i)15-s + (−0.0817 + 0.236i)16-s + (0.753 − 1.05i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.175 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.175 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.895437 + 0.750171i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.895437 + 0.750171i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.888 - 0.458i)T \) |
| 67 | \( 1 + (-8.00 - 1.70i)T \) |
| good | 3 | \( 1 + (1.91 - 0.563i)T + (2.52 - 1.62i)T^{2} \) |
| 5 | \( 1 + (-1.21 - 1.40i)T + (-0.711 + 4.94i)T^{2} \) |
| 7 | \( 1 + (-0.0846 - 1.77i)T + (-6.96 + 0.665i)T^{2} \) |
| 11 | \( 1 + (0.216 - 0.0417i)T + (10.2 - 4.08i)T^{2} \) |
| 13 | \( 1 + (-1.83 + 0.735i)T + (9.40 - 8.97i)T^{2} \) |
| 17 | \( 1 + (-3.10 + 4.36i)T + (-5.56 - 16.0i)T^{2} \) |
| 19 | \( 1 + (-0.0364 + 0.765i)T + (-18.9 - 1.80i)T^{2} \) |
| 23 | \( 1 + (1.65 + 1.57i)T + (1.09 + 22.9i)T^{2} \) |
| 29 | \( 1 + (-2.03 + 3.51i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (6.43 + 2.57i)T + (22.4 + 21.3i)T^{2} \) |
| 37 | \( 1 + (-5.49 - 9.51i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2.76 + 0.263i)T + (40.2 + 7.75i)T^{2} \) |
| 43 | \( 1 + (-4.11 - 9.02i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + (0.0177 - 0.0733i)T + (-41.7 - 21.5i)T^{2} \) |
| 53 | \( 1 + (2.52 - 5.53i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (1.38 + 9.63i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (13.9 + 2.68i)T + (56.6 + 22.6i)T^{2} \) |
| 71 | \( 1 + (9.04 + 12.6i)T + (-23.2 + 67.0i)T^{2} \) |
| 73 | \( 1 + (4.07 + 0.784i)T + (67.7 + 27.1i)T^{2} \) |
| 79 | \( 1 + (-9.91 - 7.79i)T + (18.6 + 76.7i)T^{2} \) |
| 83 | \( 1 + (-0.979 + 2.82i)T + (-65.2 - 51.3i)T^{2} \) |
| 89 | \( 1 + (5.27 + 1.54i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (3.56 + 6.16i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−13.56321170554270260508553588775, −12.31796664193610256166699399837, −11.52900127589315003973803133958, −10.65860086199845743262489416960, −9.538628010728114990057174232841, −7.938449062994021338740315078048, −6.42797445781053263904199588650, −5.79479186611567500714913814151, −4.72519531856602630021365062640, −2.81964174443770917944860679134,
1.37408470116880426492943342985, 3.84781998601382169023917366950, 5.34387732912688854878635076230, 6.01605907747246412113287408027, 7.30612987839587765740385025480, 8.970499633402325107893157353201, 10.37309264921278631271773225801, 11.05344771480419887369935713429, 12.21468892186165864388752421270, 12.81530562411163009790027243911
