| L(s) = 1 | + (0.0871 − 0.996i)2-s + (−0.984 − 0.173i)4-s + (3.66 + 1.70i)5-s + (−0.294 + 0.107i)7-s + (−0.258 + 0.965i)8-s + (2.02 − 3.49i)10-s + (−0.875 − 1.51i)11-s + (1.84 + 2.64i)13-s + (0.0811 + 0.302i)14-s + (0.939 + 0.342i)16-s + (−0.590 + 0.843i)17-s + (5.81 − 0.508i)19-s + (−3.31 − 2.31i)20-s + (−1.58 + 0.740i)22-s + (−2.42 + 0.648i)23-s + ⋯ |
| L(s) = 1 | + (0.0616 − 0.704i)2-s + (−0.492 − 0.0868i)4-s + (1.63 + 0.763i)5-s + (−0.111 + 0.0405i)7-s + (−0.0915 + 0.341i)8-s + (0.638 − 1.10i)10-s + (−0.264 − 0.457i)11-s + (0.513 + 0.732i)13-s + (0.0216 + 0.0809i)14-s + (0.234 + 0.0855i)16-s + (−0.143 + 0.204i)17-s + (1.33 − 0.116i)19-s + (−0.740 − 0.518i)20-s + (−0.338 + 0.157i)22-s + (−0.504 + 0.135i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.325i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 + 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.91028 - 0.319198i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.91028 - 0.319198i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.0871 + 0.996i)T \) |
| 3 | \( 1 \) |
| 37 | \( 1 + (-5.99 - 1.00i)T \) |
| good | 5 | \( 1 + (-3.66 - 1.70i)T + (3.21 + 3.83i)T^{2} \) |
| 7 | \( 1 + (0.294 - 0.107i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (0.875 + 1.51i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.84 - 2.64i)T + (-4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (0.590 - 0.843i)T + (-5.81 - 15.9i)T^{2} \) |
| 19 | \( 1 + (-5.81 + 0.508i)T + (18.7 - 3.29i)T^{2} \) |
| 23 | \( 1 + (2.42 - 0.648i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (2.34 + 0.627i)T + (25.1 + 14.5i)T^{2} \) |
| 31 | \( 1 + (-0.925 - 0.925i)T + 31iT^{2} \) |
| 41 | \( 1 + (0.370 - 2.10i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-4.14 + 4.14i)T - 43iT^{2} \) |
| 47 | \( 1 + (3.98 + 2.30i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.72 + 7.48i)T + (-40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-0.432 - 0.927i)T + (-37.9 + 45.1i)T^{2} \) |
| 61 | \( 1 + (-0.131 + 0.0918i)T + (20.8 - 57.3i)T^{2} \) |
| 67 | \( 1 + (5.46 + 15.0i)T + (-51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-8.74 + 10.4i)T + (-12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + 11.4iT - 73T^{2} \) |
| 79 | \( 1 + (0.448 - 0.961i)T + (-50.7 - 60.5i)T^{2} \) |
| 83 | \( 1 + (8.03 - 1.41i)T + (77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (12.5 - 5.83i)T + (57.2 - 68.1i)T^{2} \) |
| 97 | \( 1 + (-2.88 - 10.7i)T + (-84.0 + 48.5i)T^{2} \) |
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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
−10.52861349585138574679972394284, −9.566002937071762653266844124365, −9.279011333650659693398988153736, −7.956447663714018731860695509410, −6.66352434832245651120043701783, −5.95501640801334908573898821047, −5.07864224496056635922513385605, −3.56908696470741018849220506016, −2.57881610943576905509543523093, −1.53181894685241278159644444123,
1.24205680955113296409147287192, 2.73722681809699600113287617813, 4.36444808895101140052488508934, 5.49090326963033227062148009673, 5.79005995583173582093174098035, 6.92616976337739829697409956759, 7.979009846103988372610681730728, 8.863363377302984057856078577566, 9.740724222956699664857860831565, 10.06854168384367342076736374213
