L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.866 + 1.5i)3-s + (−0.499 + 0.866i)4-s + 1.73·6-s + (0.866 + 2.5i)7-s + 0.999·8-s + (−1.5 − 2.59i)9-s + (5.19 + 3i)11-s + (−0.866 − 1.49i)12-s − 1.73·13-s + (1.73 − 2i)14-s + (−0.5 − 0.866i)16-s + (−1.5 + 2.59i)18-s + (1.5 − 0.866i)19-s + (−4.5 − 0.866i)21-s − 6i·22-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.499 + 0.866i)3-s + (−0.249 + 0.433i)4-s + 0.707·6-s + (0.327 + 0.944i)7-s + 0.353·8-s + (−0.5 − 0.866i)9-s + (1.56 + 0.904i)11-s + (−0.250 − 0.433i)12-s − 0.480·13-s + (0.462 − 0.534i)14-s + (−0.125 − 0.216i)16-s + (−0.353 + 0.612i)18-s + (0.344 − 0.198i)19-s + (−0.981 − 0.188i)21-s − 1.27i·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00342 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00342 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.038441625\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.038441625\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 + (0.866 - 1.5i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.866 - 2.5i)T \) |
good | 11 | \( 1 + (-5.19 - 3i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 1.73T + 13T^{2} \) |
| 17 | \( 1 + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.5 + 0.866i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 + (3 + 1.73i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (6.06 - 3.5i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + (9 - 5.19i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (5.19 - 9i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.5 - 2.59i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-9.52 - 5.5i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 6iT - 71T^{2} \) |
| 73 | \( 1 + (-0.866 + 1.5i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 10.3iT - 83T^{2} \) |
| 89 | \( 1 + (-5.19 - 9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 15.5T + 97T^{2} \) |
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
−9.893232394665916845091933363061, −9.408180533525510186433979286764, −8.966032397772508467031330275170, −7.77402519599648092276301742038, −6.69759216791007297520150826152, −5.69315671022220113189985602189, −4.74578151945905893176970610984, −3.97480465243556634035248815015, −2.81695578373021165751490618455, −1.45718942623669486517800876856,
0.62563661543478226748887492815, 1.65871806387811295632742885332, 3.47069153867988944583461931377, 4.69104354268968871053500837952, 5.61475415208872578752253924170, 6.70454106121831505465112434962, 6.93581489965783801289761276073, 7.946268946032402637685147756489, 8.692571268675074110387367382052, 9.537376567376632004750358664235