| L(s) = 1 | + (−2.19 + 0.484i)2-s + (−0.161 + 0.986i)3-s + (2.78 − 1.28i)4-s + (0.0503 − 0.181i)5-s + (−0.121 − 2.24i)6-s + (2.06 − 1.95i)7-s + (−1.92 + 1.45i)8-s + (−0.947 − 0.319i)9-s + (−0.0229 + 0.423i)10-s + (0.782 + 0.920i)11-s + (0.821 + 2.95i)12-s + (2.31 − 0.779i)13-s + (−3.59 + 5.30i)14-s + (0.170 + 0.0790i)15-s + (−0.459 + 0.541i)16-s + (2.03 + 1.92i)17-s + ⋯ |
| L(s) = 1 | + (−1.55 + 0.342i)2-s + (−0.0934 + 0.569i)3-s + (1.39 − 0.644i)4-s + (0.0225 − 0.0811i)5-s + (−0.0497 − 0.917i)6-s + (0.781 − 0.739i)7-s + (−0.678 + 0.516i)8-s + (−0.315 − 0.106i)9-s + (−0.00725 + 0.133i)10-s + (0.235 + 0.277i)11-s + (0.237 + 0.854i)12-s + (0.641 − 0.216i)13-s + (−0.961 + 1.41i)14-s + (0.0441 + 0.0204i)15-s + (−0.114 + 0.135i)16-s + (0.493 + 0.467i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.748 - 0.663i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.748 - 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.574688 + 0.218139i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.574688 + 0.218139i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.161 - 0.986i)T \) |
| 59 | \( 1 + (-6.24 + 4.47i)T \) |
| good | 2 | \( 1 + (2.19 - 0.484i)T + (1.81 - 0.839i)T^{2} \) |
| 5 | \( 1 + (-0.0503 + 0.181i)T + (-4.28 - 2.57i)T^{2} \) |
| 7 | \( 1 + (-2.06 + 1.95i)T + (0.378 - 6.98i)T^{2} \) |
| 11 | \( 1 + (-0.782 - 0.920i)T + (-1.77 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-2.31 + 0.779i)T + (10.3 - 7.86i)T^{2} \) |
| 17 | \( 1 + (-2.03 - 1.92i)T + (0.920 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-1.85 - 4.65i)T + (-13.7 + 13.0i)T^{2} \) |
| 23 | \( 1 + (-4.29 + 0.467i)T + (22.4 - 4.94i)T^{2} \) |
| 29 | \( 1 + (3.96 + 0.871i)T + (26.3 + 12.1i)T^{2} \) |
| 31 | \( 1 + (0.590 - 1.48i)T + (-22.5 - 21.3i)T^{2} \) |
| 37 | \( 1 + (9.19 + 6.98i)T + (9.89 + 35.6i)T^{2} \) |
| 41 | \( 1 + (-7.16 - 0.778i)T + (40.0 + 8.81i)T^{2} \) |
| 43 | \( 1 + (-1.09 + 1.29i)T + (-6.95 - 42.4i)T^{2} \) |
| 47 | \( 1 + (0.710 + 2.55i)T + (-40.2 + 24.2i)T^{2} \) |
| 53 | \( 1 + (0.234 + 4.32i)T + (-52.6 + 5.73i)T^{2} \) |
| 61 | \( 1 + (9.38 - 2.06i)T + (55.3 - 25.6i)T^{2} \) |
| 67 | \( 1 + (-9.39 + 7.14i)T + (17.9 - 64.5i)T^{2} \) |
| 71 | \( 1 + (-0.898 - 3.23i)T + (-60.8 + 36.6i)T^{2} \) |
| 73 | \( 1 + (3.67 - 5.42i)T + (-27.0 - 67.8i)T^{2} \) |
| 79 | \( 1 + (-1.22 - 7.49i)T + (-74.8 + 25.2i)T^{2} \) |
| 83 | \( 1 + (-3.10 - 5.85i)T + (-46.5 + 68.6i)T^{2} \) |
| 89 | \( 1 + (10.5 + 2.32i)T + (80.7 + 37.3i)T^{2} \) |
| 97 | \( 1 + (5.49 + 8.09i)T + (-35.9 + 90.1i)T^{2} \) |
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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
−12.62699303889894021667110620933, −11.14932929598395490460173928478, −10.68395250011361183753247667793, −9.730567928039672788523083326949, −8.779617228614784384244047255575, −7.894032379629840171361367418226, −6.95734306136609629802215880783, −5.46911683592100192439269369713, −3.86896037583963542515276697118, −1.36190225884384503586961476873,
1.25860982439408514904464590270, 2.74216049539127728557920401673, 5.17732183553574105274585839214, 6.70977217436074241746954557838, 7.70722652267199884889946117952, 8.707792734465999484329130336876, 9.252186221077753538789313734764, 10.69582584131241714384315640898, 11.38110845276107989050658887314, 12.09366684526573985787742405421
