| L(s) = 1 | + (−0.740 + 0.871i)2-s + (−1.61 + 0.629i)3-s + (0.111 + 0.682i)4-s + (−1.53 + 0.813i)5-s + (0.646 − 1.87i)6-s + (0.0293 + 0.00319i)7-s + (−2.63 − 1.58i)8-s + (2.20 − 2.03i)9-s + (0.427 − 1.94i)10-s + (0.0537 + 0.0181i)11-s + (−0.610 − 1.03i)12-s + (−5.37 − 1.49i)13-s + (−0.0245 + 0.0232i)14-s + (1.96 − 2.27i)15-s + (2.02 − 0.682i)16-s + (−0.278 − 2.56i)17-s + ⋯ |
| L(s) = 1 | + (−0.523 + 0.616i)2-s + (−0.931 + 0.363i)3-s + (0.0559 + 0.341i)4-s + (−0.686 + 0.363i)5-s + (0.263 − 0.764i)6-s + (0.0110 + 0.00120i)7-s + (−0.932 − 0.561i)8-s + (0.735 − 0.677i)9-s + (0.135 − 0.613i)10-s + (0.0161 + 0.00545i)11-s + (−0.176 − 0.297i)12-s + (−1.48 − 0.413i)13-s + (−0.00655 + 0.00620i)14-s + (0.507 − 0.588i)15-s + (0.506 − 0.170i)16-s + (−0.0675 − 0.621i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.743 + 0.668i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.743 + 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0721392 - 0.188272i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0721392 - 0.188272i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.61 - 0.629i)T \) |
| 59 | \( 1 + (-1.31 - 7.56i)T \) |
| good | 2 | \( 1 + (0.740 - 0.871i)T + (-0.323 - 1.97i)T^{2} \) |
| 5 | \( 1 + (1.53 - 0.813i)T + (2.80 - 4.13i)T^{2} \) |
| 7 | \( 1 + (-0.0293 - 0.00319i)T + (6.83 + 1.50i)T^{2} \) |
| 11 | \( 1 + (-0.0537 - 0.0181i)T + (8.75 + 6.65i)T^{2} \) |
| 13 | \( 1 + (5.37 + 1.49i)T + (11.1 + 6.70i)T^{2} \) |
| 17 | \( 1 + (0.278 + 2.56i)T + (-16.6 + 3.65i)T^{2} \) |
| 19 | \( 1 + (0.163 - 3.00i)T + (-18.8 - 2.05i)T^{2} \) |
| 23 | \( 1 + (2.31 - 1.07i)T + (14.8 - 17.5i)T^{2} \) |
| 29 | \( 1 + (2.59 - 2.20i)T + (4.69 - 28.6i)T^{2} \) |
| 31 | \( 1 + (-1.74 + 0.0947i)T + (30.8 - 3.35i)T^{2} \) |
| 37 | \( 1 + (-1.62 - 2.70i)T + (-17.3 + 32.6i)T^{2} \) |
| 41 | \( 1 + (4.30 - 9.30i)T + (-26.5 - 31.2i)T^{2} \) |
| 43 | \( 1 + (-2.60 - 7.73i)T + (-34.2 + 26.0i)T^{2} \) |
| 47 | \( 1 + (-2.26 + 4.27i)T + (-26.3 - 38.9i)T^{2} \) |
| 53 | \( 1 + (0.0348 + 0.158i)T + (-48.1 + 22.2i)T^{2} \) |
| 61 | \( 1 + (8.42 + 7.15i)T + (9.86 + 60.1i)T^{2} \) |
| 67 | \( 1 + (4.87 - 8.09i)T + (-31.3 - 59.1i)T^{2} \) |
| 71 | \( 1 + (-4.79 - 2.54i)T + (39.8 + 58.7i)T^{2} \) |
| 73 | \( 1 + (9.22 + 9.73i)T + (-3.95 + 72.8i)T^{2} \) |
| 79 | \( 1 + (-13.0 + 9.93i)T + (21.1 - 76.1i)T^{2} \) |
| 83 | \( 1 + (2.73 + 6.85i)T + (-60.2 + 57.0i)T^{2} \) |
| 89 | \( 1 + (-1.93 - 2.27i)T + (-14.3 + 87.8i)T^{2} \) |
| 97 | \( 1 + (-4.46 + 4.70i)T + (-5.25 - 96.8i)T^{2} \) |
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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
−12.99698631595595784193912093439, −12.02909886545178970323290911362, −11.53057401497393595089114629700, −10.18126031329867135816183934552, −9.383875758341479532093264888050, −7.87503024841323761007997764970, −7.24450667922834272166934723176, −6.13184343310726593744426499961, −4.73763131580668158015753977417, −3.30281251265418268068799741164,
0.23202100405809815146354076925, 2.12398343180283751317249282308, 4.42420468938372938670631158711, 5.58220097004177554598759742656, 6.83664571207840645890707217188, 7.991387839217391081834812944271, 9.303448598298691600989200896260, 10.28010645496132849627862177334, 11.13356323574800432630215488018, 12.04048618483491577218444811393
