L(s) = 1 | − i·2-s + (1.05 − 1.37i)3-s − 4-s − 5-s + (−1.37 − 1.05i)6-s + i·8-s + (−0.772 − 2.89i)9-s + i·10-s − 1.09i·11-s + (−1.05 + 1.37i)12-s − 3.66i·13-s + (−1.05 + 1.37i)15-s + 16-s + 4.39·17-s + (−2.89 + 0.772i)18-s − 4.59i·19-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (0.609 − 0.792i)3-s − 0.5·4-s − 0.447·5-s + (−0.560 − 0.430i)6-s + 0.353i·8-s + (−0.257 − 0.966i)9-s + 0.316i·10-s − 0.331i·11-s + (−0.304 + 0.396i)12-s − 1.01i·13-s + (−0.272 + 0.354i)15-s + 0.250·16-s + 1.06·17-s + (−0.683 + 0.182i)18-s − 1.05i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 - 0.231i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.972 - 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.342634319\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.342634319\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + (-1.05 + 1.37i)T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 1.09iT - 11T^{2} \) |
| 13 | \( 1 + 3.66iT - 13T^{2} \) |
| 17 | \( 1 - 4.39T + 17T^{2} \) |
| 19 | \( 1 + 4.59iT - 19T^{2} \) |
| 23 | \( 1 + 1.51iT - 23T^{2} \) |
| 29 | \( 1 - 1.85iT - 29T^{2} \) |
| 31 | \( 1 - 4.34iT - 31T^{2} \) |
| 37 | \( 1 + 11.7T + 37T^{2} \) |
| 41 | \( 1 + 6.08T + 41T^{2} \) |
| 43 | \( 1 + 1.89T + 43T^{2} \) |
| 47 | \( 1 + 5.13T + 47T^{2} \) |
| 53 | \( 1 - 11.5iT - 53T^{2} \) |
| 59 | \( 1 - 8.82T + 59T^{2} \) |
| 61 | \( 1 + 11.4iT - 61T^{2} \) |
| 67 | \( 1 + 12.2T + 67T^{2} \) |
| 71 | \( 1 + 2.24iT - 71T^{2} \) |
| 73 | \( 1 + 6.69iT - 73T^{2} \) |
| 79 | \( 1 + 14.9T + 79T^{2} \) |
| 83 | \( 1 - 15.3T + 83T^{2} \) |
| 89 | \( 1 + 1.22T + 89T^{2} \) |
| 97 | \( 1 + 8.60iT - 97T^{2} \) |
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
−8.865266297207261997419347929108, −8.436397516271187219508468299167, −7.58377445605712178701273823774, −6.85308222797367614142881072277, −5.71375998797022019234556158400, −4.77316539575636064153060962216, −3.33649766320576440471312549032, −3.10126060543766776844718480443, −1.68732874308336656437607087195, −0.49776688382893755811103114975,
1.83922147768863321234121275955, 3.37829902985457659911464796034, 3.98728277347190655467361099378, 4.92311285865344680793759254646, 5.70752222011185370174694262842, 6.86294238320290204754733685593, 7.63301691407945536970916005218, 8.343614160099109536380845039131, 8.985203413668787705157365230651, 9.949523033161295770370278759216