| L(s) = 1 | + (0.415 + 0.909i)3-s + (0.735 + 0.848i)5-s + (0.891 + 0.572i)7-s + (−0.654 + 0.755i)9-s + (0.347 + 2.41i)11-s + (1.00 − 0.643i)13-s + (−0.466 + 1.02i)15-s + (−0.810 + 0.237i)17-s + (1.83 + 0.538i)19-s + (−0.150 + 1.04i)21-s + (−0.265 + 4.78i)23-s + (0.531 − 3.70i)25-s + (−0.959 − 0.281i)27-s + (−0.551 + 0.161i)29-s + (0.110 − 0.242i)31-s + ⋯ |
| L(s) = 1 | + (0.239 + 0.525i)3-s + (0.328 + 0.379i)5-s + (0.336 + 0.216i)7-s + (−0.218 + 0.251i)9-s + (0.104 + 0.729i)11-s + (0.277 − 0.178i)13-s + (−0.120 + 0.263i)15-s + (−0.196 + 0.0577i)17-s + (0.420 + 0.123i)19-s + (−0.0329 + 0.228i)21-s + (−0.0553 + 0.998i)23-s + (0.106 − 0.740i)25-s + (−0.184 − 0.0542i)27-s + (−0.102 + 0.0300i)29-s + (0.0198 − 0.0435i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.556 - 0.831i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.556 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.29398 + 0.691039i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.29398 + 0.691039i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.415 - 0.909i)T \) |
| 23 | \( 1 + (0.265 - 4.78i)T \) |
| good | 5 | \( 1 + (-0.735 - 0.848i)T + (-0.711 + 4.94i)T^{2} \) |
| 7 | \( 1 + (-0.891 - 0.572i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (-0.347 - 2.41i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (-1.00 + 0.643i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (0.810 - 0.237i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (-1.83 - 0.538i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (0.551 - 0.161i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-0.110 + 0.242i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (-5.68 + 6.55i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (3.19 + 3.68i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (1.36 + 2.99i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 0.561T + 47T^{2} \) |
| 53 | \( 1 + (5.21 + 3.35i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (1.23 - 0.794i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-3.69 + 8.08i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (-0.709 + 4.93i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (0.0399 - 0.277i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (4.93 + 1.45i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (4.92 - 3.16i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-3.38 + 3.90i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (5.04 + 11.0i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (-2.64 - 3.04i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−11.91233825951305872476370906456, −11.00783625693999087834001649720, −10.07383900947204043540000730117, −9.329447955833342477781443145191, −8.243447833607067550456992441944, −7.19081067819334577605292055884, −5.93757543693229476580449466447, −4.83663804687105997124512935159, −3.56383877396662298521373484230, −2.10139871849701340085016270705,
1.30260894792712148541174146905, 2.96851560215970365403800198607, 4.50518475632505997372369032758, 5.78457964396976599260938025014, 6.79032659361103654866558723272, 7.972258970281469955467478258246, 8.752745725167957323509411146337, 9.718307403081292361361675467117, 10.96052162927975006459025366031, 11.70750616437205456911765915931
