| L(s) = 1 | + (0.988 − 0.149i)2-s + (−1.46 − 1.00i)3-s + (0.955 − 0.294i)4-s + (0.169 − 2.25i)5-s + (−1.60 − 0.771i)6-s + (2.40 + 1.10i)7-s + (0.900 − 0.433i)8-s + (0.0601 + 0.153i)9-s + (−0.169 − 2.25i)10-s + (−2.21 + 5.65i)11-s + (−1.69 − 0.524i)12-s + (0.113 + 0.142i)13-s + (2.54 + 0.729i)14-s + (−2.50 + 3.14i)15-s + (0.826 − 0.563i)16-s + (−3.61 + 3.35i)17-s + ⋯ |
| L(s) = 1 | + (0.699 − 0.105i)2-s + (−0.848 − 0.578i)3-s + (0.477 − 0.147i)4-s + (0.0755 − 1.00i)5-s + (−0.654 − 0.315i)6-s + (0.909 + 0.415i)7-s + (0.318 − 0.153i)8-s + (0.0200 + 0.0511i)9-s + (−0.0534 − 0.713i)10-s + (−0.669 + 1.70i)11-s + (−0.490 − 0.151i)12-s + (0.0315 + 0.0396i)13-s + (0.679 + 0.194i)14-s + (−0.647 + 0.812i)15-s + (0.206 − 0.140i)16-s + (−0.876 + 0.813i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.637 + 0.770i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.08138 - 0.508663i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.08138 - 0.508663i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.988 + 0.149i)T \) |
| 7 | \( 1 + (-2.40 - 1.10i)T \) |
| good | 3 | \( 1 + (1.46 + 1.00i)T + (1.09 + 2.79i)T^{2} \) |
| 5 | \( 1 + (-0.169 + 2.25i)T + (-4.94 - 0.745i)T^{2} \) |
| 11 | \( 1 + (2.21 - 5.65i)T + (-8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (-0.113 - 0.142i)T + (-2.89 + 12.6i)T^{2} \) |
| 17 | \( 1 + (3.61 - 3.35i)T + (1.27 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-1.31 + 2.27i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.05 - 2.83i)T + (1.71 + 22.9i)T^{2} \) |
| 29 | \( 1 + (2.05 + 9.02i)T + (-26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (-0.668 - 1.15i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.71 - 0.838i)T + (30.5 + 20.8i)T^{2} \) |
| 41 | \( 1 + (0.529 - 0.255i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (7.91 + 3.81i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (8.72 - 1.31i)T + (44.9 - 13.8i)T^{2} \) |
| 53 | \( 1 + (-0.187 + 0.0577i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (0.873 + 11.6i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (0.523 + 0.161i)T + (50.4 + 34.3i)T^{2} \) |
| 67 | \( 1 + (3.83 + 6.65i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.01 + 4.43i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-7.77 - 1.17i)T + (69.7 + 21.5i)T^{2} \) |
| 79 | \( 1 + (-0.263 + 0.455i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (11.1 - 13.9i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (-1.42 - 3.62i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + 4.21T + 97T^{2} \) |
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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
−13.33323815707852881037197629323, −12.69526673280448668589458185251, −11.91286520296746140073740190026, −11.05639925946563022591718484559, −9.498101296121498290910802161293, −8.023709924547199456468190292995, −6.72460224850930630923053169942, −5.33306653761653061474000378989, −4.61379085755875160002566026777, −1.83527280729847770990177835442,
3.06158043789261951888781983266, 4.75874305409681743692944509502, 5.74614948603747473146251465931, 7.01278011114903077797152053243, 8.388841630709497045434015457256, 10.42016709869119614270032900064, 11.04929190661923302481650248339, 11.51248064958105090319687118299, 13.27638000860000586878224188370, 14.11360635546730158668836879959
