L(s) = 1 | + (0.260 + 0.801i)3-s + (0.809 + 0.587i)5-s + (1.46 − 4.52i)7-s + (1.85 − 1.34i)9-s + (0.231 − 3.30i)11-s + (−3.06 + 2.22i)13-s + (−0.260 + 0.801i)15-s + (−4.03 − 2.92i)17-s + (0.964 + 2.96i)19-s + 4.01·21-s + 9.39·23-s + (0.309 + 0.951i)25-s + (3.60 + 2.62i)27-s + (−0.805 + 2.47i)29-s + (2.98 − 2.16i)31-s + ⋯ |
L(s) = 1 | + (0.150 + 0.462i)3-s + (0.361 + 0.262i)5-s + (0.555 − 1.70i)7-s + (0.617 − 0.448i)9-s + (0.0699 − 0.997i)11-s + (−0.848 + 0.616i)13-s + (−0.0672 + 0.207i)15-s + (−0.977 − 0.710i)17-s + (0.221 + 0.680i)19-s + 0.875·21-s + 1.95·23-s + (0.0618 + 0.190i)25-s + (0.694 + 0.504i)27-s + (−0.149 + 0.460i)29-s + (0.535 − 0.389i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 + 0.396i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.917 + 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.61079 - 0.333153i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.61079 - 0.333153i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.809 - 0.587i)T \) |
| 11 | \( 1 + (-0.231 + 3.30i)T \) |
good | 3 | \( 1 + (-0.260 - 0.801i)T + (-2.42 + 1.76i)T^{2} \) |
| 7 | \( 1 + (-1.46 + 4.52i)T + (-5.66 - 4.11i)T^{2} \) |
| 13 | \( 1 + (3.06 - 2.22i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (4.03 + 2.92i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-0.964 - 2.96i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 - 9.39T + 23T^{2} \) |
| 29 | \( 1 + (0.805 - 2.47i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-2.98 + 2.16i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (1.06 - 3.27i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-2.88 - 8.87i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 11.7T + 43T^{2} \) |
| 47 | \( 1 + (0.513 + 1.58i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-1.56 + 1.14i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-0.0321 + 0.0988i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-8.21 - 5.96i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + 4.99T + 67T^{2} \) |
| 71 | \( 1 + (-5.03 - 3.66i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (2.25 - 6.94i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (8.53 - 6.19i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (4.60 + 3.34i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 - 13.6T + 89T^{2} \) |
| 97 | \( 1 + (11.0 - 8.05i)T + (29.9 - 92.2i)T^{2} \) |
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
−10.99927872071705980327985929443, −10.14103625552366521954125452856, −9.481515961132192877256714374362, −8.389443055749256609410829368577, −7.11549226418797774091726415108, −6.74505432467294661766939500486, −5.02085606002464017516420736220, −4.23746134367672528702105753216, −3.11902677859101332690357483945, −1.18190955653075260468001455583,
1.84183858800700416443801605238, 2.61550841554823985253451451127, 4.74633720825020161014496870072, 5.25220305140558457874910320355, 6.60274674011599696579628020247, 7.50221339341847800883488368767, 8.572813739001788176607386969485, 9.205402941007274873828311338060, 10.22317725684370155795346554401, 11.28370545379209527719669889369