| L(s) = 1 | + (−1.05 + 0.224i)2-s + (1.71 + 0.259i)3-s + (−0.766 + 0.341i)4-s + (−1.13 − 1.92i)5-s + (−1.86 + 0.110i)6-s + (0.316 + 0.548i)7-s + (2.47 − 1.79i)8-s + (2.86 + 0.888i)9-s + (1.63 + 1.77i)10-s + (5.10 − 1.08i)11-s + (−1.40 + 0.385i)12-s + (3.72 + 0.791i)13-s + (−0.456 − 0.506i)14-s + (−1.44 − 3.59i)15-s + (−1.08 + 1.20i)16-s + (−0.365 + 0.265i)17-s + ⋯ |
| L(s) = 1 | + (−0.745 + 0.158i)2-s + (0.988 + 0.149i)3-s + (−0.383 + 0.170i)4-s + (−0.509 − 0.860i)5-s + (−0.760 + 0.0449i)6-s + (0.119 + 0.207i)7-s + (0.874 − 0.635i)8-s + (0.955 + 0.296i)9-s + (0.515 + 0.560i)10-s + (1.53 − 0.327i)11-s + (−0.404 + 0.111i)12-s + (1.03 + 0.219i)13-s + (−0.121 − 0.135i)14-s + (−0.374 − 0.927i)15-s + (−0.270 + 0.300i)16-s + (−0.0886 + 0.0643i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0413i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0413i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.06294 + 0.0219952i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.06294 + 0.0219952i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.71 - 0.259i)T \) |
| 5 | \( 1 + (1.13 + 1.92i)T \) |
| good | 2 | \( 1 + (1.05 - 0.224i)T + (1.82 - 0.813i)T^{2} \) |
| 7 | \( 1 + (-0.316 - 0.548i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-5.10 + 1.08i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (-3.72 - 0.791i)T + (11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (0.365 - 0.265i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (3.97 - 2.88i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (0.0686 + 0.0762i)T + (-2.40 + 22.8i)T^{2} \) |
| 29 | \( 1 + (-1.04 + 9.98i)T + (-28.3 - 6.02i)T^{2} \) |
| 31 | \( 1 + (0.422 + 4.01i)T + (-30.3 + 6.44i)T^{2} \) |
| 37 | \( 1 + (-0.0800 - 0.246i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (7.65 + 1.62i)T + (37.4 + 16.6i)T^{2} \) |
| 43 | \( 1 + (-3.68 - 6.38i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (1.10 - 10.5i)T + (-45.9 - 9.77i)T^{2} \) |
| 53 | \( 1 + (4.56 + 3.31i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (7.00 + 1.48i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (4.50 - 0.958i)T + (55.7 - 24.8i)T^{2} \) |
| 67 | \( 1 + (0.364 + 3.46i)T + (-65.5 + 13.9i)T^{2} \) |
| 71 | \( 1 + (3.47 + 2.52i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (4.75 - 14.6i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (0.142 - 1.35i)T + (-77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (-1.95 - 0.870i)T + (55.5 + 61.6i)T^{2} \) |
| 89 | \( 1 + (5.42 - 16.7i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-0.637 + 6.06i)T + (-94.8 - 20.1i)T^{2} \) |
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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
−12.35282211864527302425903378522, −11.22703283085274450903246084890, −9.807581335527809246580081166054, −9.082818217091756030860182598716, −8.443632295778827708962925828808, −7.82949343407203679442433502010, −6.35079011657511941187167871416, −4.34876772698173677475504821687, −3.81885573122957050813747212555, −1.41366533242909221419533272451,
1.57177209277507770569990350041, 3.41353622791173039220012040174, 4.43840697375891569258312741273, 6.56959215502592850525230007684, 7.39303263726417167351484478073, 8.655313692829266957318264885457, 8.985773024326805921310094533339, 10.28859539592173154442397949998, 10.91961526474529891791269219797, 12.15441937630736051034893804364
