| L(s) = 1 | + (−0.142 − 0.989i)2-s + (0.415 − 0.909i)3-s + (−0.959 + 0.281i)4-s + (0.654 − 0.755i)5-s + (−0.959 − 0.281i)6-s + (2.31 − 1.48i)7-s + (0.415 + 0.909i)8-s + (−0.654 − 0.755i)9-s + (−0.841 − 0.540i)10-s + (0.0200 − 0.139i)11-s + (−0.142 + 0.989i)12-s + (−3.76 − 2.42i)13-s + (−1.80 − 2.07i)14-s + (−0.415 − 0.909i)15-s + (0.841 − 0.540i)16-s + (1.08 + 0.318i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 − 0.699i)2-s + (0.239 − 0.525i)3-s + (−0.479 + 0.140i)4-s + (0.292 − 0.337i)5-s + (−0.391 − 0.115i)6-s + (0.874 − 0.561i)7-s + (0.146 + 0.321i)8-s + (−0.218 − 0.251i)9-s + (−0.266 − 0.170i)10-s + (0.00603 − 0.0419i)11-s + (−0.0410 + 0.285i)12-s + (−1.04 − 0.671i)13-s + (−0.481 − 0.555i)14-s + (−0.107 − 0.234i)15-s + (0.210 − 0.135i)16-s + (0.263 + 0.0772i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.746 + 0.665i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.547901 - 1.43879i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.547901 - 1.43879i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.142 + 0.989i)T \) |
| 3 | \( 1 + (-0.415 + 0.909i)T \) |
| 5 | \( 1 + (-0.654 + 0.755i)T \) |
| 23 | \( 1 + (3.77 + 2.95i)T \) |
| good | 7 | \( 1 + (-2.31 + 1.48i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (-0.0200 + 0.139i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (3.76 + 2.42i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (-1.08 - 0.318i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (-4.66 + 1.37i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (-9.35 - 2.74i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (1.52 + 3.33i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (4.25 + 4.90i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (7.94 - 9.16i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (-2.22 + 4.88i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + 4.61T + 47T^{2} \) |
| 53 | \( 1 + (0.688 - 0.442i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (2.53 + 1.63i)T + (24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (4.87 + 10.6i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (-1.15 - 8.02i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (-1.03 - 7.19i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-8.86 + 2.60i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (-10.3 - 6.66i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (-10.5 - 12.1i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-2.26 + 4.96i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (-6.94 + 8.01i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−10.15068330924419649722939383770, −9.432713424104789107709274135243, −8.270733075907775130930562587597, −7.82418404161924092215845372113, −6.74752949329489742714618745712, −5.34274829008173171921371044208, −4.62361758504803204491511079333, −3.24667329754728095767005371131, −2.08600947254292948473093363880, −0.858439599755470623206362895117,
1.90620578185248865346761045447, 3.30177395821210730853761444833, 4.71343081834093605820766512401, 5.25888111015339421709300980777, 6.36754655245920777961276779635, 7.41898424167487201593465298244, 8.162369427693345389302868162099, 9.062952969945016346162766790427, 9.831498891491205939527399567135, 10.49028572748319056724057654089
