| L(s) = 1 | + (−0.142 − 0.989i)2-s + (−1.16 − 1.28i)3-s + (−0.959 + 0.281i)4-s + (2.23 − 0.0433i)5-s + (−1.10 + 1.33i)6-s + (1.95 − 1.25i)7-s + (0.415 + 0.909i)8-s + (−0.295 + 2.98i)9-s + (−0.361 − 2.20i)10-s + (−0.0412 + 0.286i)11-s + (1.47 + 0.904i)12-s + (1.87 − 2.91i)13-s + (−1.52 − 1.75i)14-s + (−2.65 − 2.81i)15-s + (0.841 − 0.540i)16-s + (1.45 − 4.96i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 − 0.699i)2-s + (−0.671 − 0.741i)3-s + (−0.479 + 0.140i)4-s + (0.999 − 0.0193i)5-s + (−0.451 + 0.544i)6-s + (0.740 − 0.475i)7-s + (0.146 + 0.321i)8-s + (−0.0985 + 0.995i)9-s + (−0.114 − 0.697i)10-s + (−0.0124 + 0.0864i)11-s + (0.426 + 0.260i)12-s + (0.519 − 0.807i)13-s + (−0.407 − 0.470i)14-s + (−0.685 − 0.727i)15-s + (0.210 − 0.135i)16-s + (0.353 − 1.20i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.393 + 0.919i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.393 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.789420 - 1.19661i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.789420 - 1.19661i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.16 + 1.28i)T \) |
| 5 | \( 1 + (-2.23 + 0.0433i)T \) |
| 23 | \( 1 + (-4.72 - 0.830i)T \) |
| good | 7 | \( 1 + (-1.95 + 1.25i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (0.0412 - 0.286i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (-1.87 + 2.91i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-1.45 + 4.96i)T + (-14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (-2.20 - 7.52i)T + (-15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (-1.28 + 4.39i)T + (-24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (0.269 + 0.589i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (6.06 + 6.99i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (6.02 + 5.22i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (4.06 - 8.89i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + 12.5T + 47T^{2} \) |
| 53 | \( 1 + (5.71 + 8.89i)T + (-22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-1.03 + 1.61i)T + (-24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-5.15 + 2.35i)T + (39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (-1.48 - 10.3i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (-8.25 + 1.18i)T + (68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-3.93 - 13.4i)T + (-61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-5.89 + 9.17i)T + (-32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-5.21 + 4.51i)T + (11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (2.75 - 6.03i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (-10.7 + 12.4i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−10.27231193952968107360393828545, −9.688307202076444530768604152982, −8.367885008057678099036456168656, −7.68098299496944438074113082504, −6.60964614913652047228114847949, −5.48067435719466286288922627614, −4.98590331718152181401622306448, −3.32672030439153955804882845070, −1.93779126076892135108185484064, −1.00930948542926428409796270008,
1.48896516382611606864359442603, 3.33470023977120228247555589945, 4.97347695688937795840948720578, 5.07116758154515854190191991104, 6.36706823261443682142211077826, 6.80161881742281596432620258134, 8.455731474948164453748331191603, 8.966804822821251347241800870530, 9.779096024604467017311277701193, 10.69750542003271561964188760024
