L(s) = 1 | + (1 + 1.73i)3-s + (0.5 − 0.866i)5-s + (1.13 − 2.38i)7-s + (−0.499 + 0.866i)9-s + (−1.13 − 1.97i)11-s + 2·13-s + 1.99·15-s + (−3.63 − 6.30i)17-s + (0.5 − 0.866i)19-s + (5.27 − 0.418i)21-s + (3.77 − 6.53i)23-s + (2 + 3.46i)25-s + 4.00·27-s − 4·29-s + (2 + 3.46i)31-s + ⋯ |
L(s) = 1 | + (0.577 + 0.999i)3-s + (0.223 − 0.387i)5-s + (0.429 − 0.902i)7-s + (−0.166 + 0.288i)9-s + (−0.342 − 0.594i)11-s + 0.554·13-s + 0.516·15-s + (−0.882 − 1.52i)17-s + (0.114 − 0.198i)19-s + (1.15 − 0.0913i)21-s + (0.787 − 1.36i)23-s + (0.400 + 0.692i)25-s + 0.769·27-s − 0.742·29-s + (0.359 + 0.622i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1064 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.371i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1064 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.928 + 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.084941264\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.084941264\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-1.13 + 2.38i)T \) |
| 19 | \( 1 + (-0.5 + 0.866i)T \) |
good | 3 | \( 1 + (-1 - 1.73i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.5 + 0.866i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.13 + 1.97i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + (3.63 + 6.30i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-3.77 + 6.53i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 4.54T + 41T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 + (-1.13 + 1.97i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.27 - 5.67i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (6.13 - 10.6i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 + (1.86 + 3.22i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.274 - 0.476i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 11.5T + 83T^{2} \) |
| 89 | \( 1 + (8.27 - 14.3i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 8.54T + 97T^{2} \) |
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
−9.787603877037124753149550073231, −8.919114582488040928971965732452, −8.594645967354352885021259570331, −7.37550858480586501013252029128, −6.58371535549691392393358800706, −5.14123563982487800681224342904, −4.63879686091347984091212209607, −3.64318409894071543183423344427, −2.69933173894425796717980523193, −0.928192953140979187610341207467,
1.67923524593089643365672993965, 2.24860615017460649403314278932, 3.45661572302742982376228686599, 4.80636586941462413634319296665, 5.88374779516592086345983494921, 6.64263002255482835675462554988, 7.53921933280499869396762476310, 8.272481267488446567658373671126, 8.861990321745235290632223200273, 9.878091536541258072277823929412