| 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
−10.49028572748319056724057654089, −9.831498891491205939527399567135, −9.062952969945016346162766790427, −8.162369427693345389302868162099, −7.41898424167487201593465298244, −6.36754655245920777961276779635, −5.25888111015339421709300980777, −4.71343081834093605820766512401, −3.30177395821210730853761444833, −1.90620578185248865346761045447,
0.858439599755470623206362895117, 2.08600947254292948473093363880, 3.24667329754728095767005371131, 4.62361758504803204491511079333, 5.34274829008173171921371044208, 6.74752949329489742714618745712, 7.82418404161924092215845372113, 8.270733075907775130930562587597, 9.432713424104789107709274135243, 10.15068330924419649722939383770
