| L(s) = 1 | + (0.561 + 1.29i)2-s + (0.400 − 0.550i)3-s + (−1.36 + 1.45i)4-s + (−2.02 − 0.946i)5-s + (0.939 + 0.210i)6-s + (−3.10 − 3.10i)7-s + (−2.66 − 0.959i)8-s + (0.783 + 2.41i)9-s + (0.0918 − 3.16i)10-s + (−4.42 + 2.25i)11-s + (0.254 + 1.33i)12-s + (0.544 + 1.67i)13-s + (2.28 − 5.76i)14-s + (−1.33 + 0.736i)15-s + (−0.248 − 3.99i)16-s + (−6.94 − 1.09i)17-s + ⋯ |
| L(s) = 1 | + (0.396 + 0.917i)2-s + (0.230 − 0.317i)3-s + (−0.684 + 0.728i)4-s + (−0.905 − 0.423i)5-s + (0.383 + 0.0857i)6-s + (−1.17 − 1.17i)7-s + (−0.940 − 0.339i)8-s + (0.261 + 0.804i)9-s + (0.0290 − 0.999i)10-s + (−1.33 + 0.679i)11-s + (0.0735 + 0.386i)12-s + (0.151 + 0.465i)13-s + (0.610 − 1.54i)14-s + (−0.343 + 0.190i)15-s + (−0.0621 − 0.998i)16-s + (−1.68 − 0.266i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.715 + 0.699i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.715 + 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00759667 - 0.0186383i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00759667 - 0.0186383i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.561 - 1.29i)T \) |
| 5 | \( 1 + (2.02 + 0.946i)T \) |
| good | 3 | \( 1 + (-0.400 + 0.550i)T + (-0.927 - 2.85i)T^{2} \) |
| 7 | \( 1 + (3.10 + 3.10i)T + 7iT^{2} \) |
| 11 | \( 1 + (4.42 - 2.25i)T + (6.46 - 8.89i)T^{2} \) |
| 13 | \( 1 + (-0.544 - 1.67i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (6.94 + 1.09i)T + (16.1 + 5.25i)T^{2} \) |
| 19 | \( 1 + (-0.811 - 0.128i)T + (18.0 + 5.87i)T^{2} \) |
| 23 | \( 1 + (1.41 + 2.78i)T + (-13.5 + 18.6i)T^{2} \) |
| 29 | \( 1 + (-8.71 + 1.38i)T + (27.5 - 8.96i)T^{2} \) |
| 31 | \( 1 + (0.119 + 0.163i)T + (-9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (0.280 + 0.864i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (3.50 - 1.13i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 5.05T + 43T^{2} \) |
| 47 | \( 1 + (6.06 - 0.961i)T + (44.6 - 14.5i)T^{2} \) |
| 53 | \( 1 + (-0.687 + 0.945i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (9.62 + 4.90i)T + (34.6 + 47.7i)T^{2} \) |
| 61 | \( 1 + (8.45 - 4.30i)T + (35.8 - 49.3i)T^{2} \) |
| 67 | \( 1 + (-5.99 + 4.35i)T + (20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (-0.936 - 0.680i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (10.7 - 5.50i)T + (42.9 - 59.0i)T^{2} \) |
| 79 | \( 1 + (11.4 + 8.29i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (5.37 + 7.40i)T + (-25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (2.37 - 7.30i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-0.344 - 2.17i)T + (-92.2 + 29.9i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.22712788778662882240389984386, −10.87166174142313165493001766773, −9.956671538897324969422328437512, −8.756869572451989973156457231505, −7.86340529620299276614723918017, −7.19680559954774164822631668114, −6.50393293727718666716920930406, −4.77158002722802855571328035014, −4.30125808345740418518653708469, −2.86260341145239483924116353366,
0.01061622610507419194454085013, 2.76603673362162745332279306295, 3.22630994839985917007838888975, 4.46206536750557283060247199313, 5.78049442477003660400077609089, 6.65709020775857081774605074519, 8.348513547180332099487502372589, 8.971404224137067907998716961002, 9.996731467957134230770346563762, 10.74209079405872990274745315237
