L(s) = 1 | − 1.41·2-s + (1.41 + 2.64i)3-s + 2.00·4-s + (−2.00 − 3.74i)6-s + 2.64i·7-s − 2.82·8-s + (−5 + 7.48i)9-s + 0.412i·11-s + (2.82 + 5.29i)12-s − 20.5i·13-s − 3.74i·14-s + 4.00·16-s − 15.8·17-s + (7.07 − 10.5i)18-s + 16·19-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.471 + 0.881i)3-s + 0.500·4-s + (−0.333 − 0.623i)6-s + 0.377i·7-s − 0.353·8-s + (−0.555 + 0.831i)9-s + 0.0374i·11-s + (0.235 + 0.440i)12-s − 1.58i·13-s − 0.267i·14-s + 0.250·16-s − 0.934·17-s + (0.392 − 0.587i)18-s + 0.842·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0272 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0272 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5441654483\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5441654483\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 + (-1.41 - 2.64i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 2.64iT \) |
good | 11 | \( 1 - 0.412iT - 121T^{2} \) |
| 13 | \( 1 + 20.5iT - 169T^{2} \) |
| 17 | \( 1 + 15.8T + 289T^{2} \) |
| 19 | \( 1 - 16T + 361T^{2} \) |
| 23 | \( 1 + 36.0T + 529T^{2} \) |
| 29 | \( 1 + 20.8iT - 841T^{2} \) |
| 31 | \( 1 + 5.54T + 961T^{2} \) |
| 37 | \( 1 - 20iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 76.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 51.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 8.48T + 2.20e3T^{2} \) |
| 53 | \( 1 + 50.9T + 2.80e3T^{2} \) |
| 59 | \( 1 - 1.64iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 66.9T + 3.72e3T^{2} \) |
| 67 | \( 1 - 49.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 87.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 12.3iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 84.9T + 6.24e3T^{2} \) |
| 83 | \( 1 - 4.12T + 6.88e3T^{2} \) |
| 89 | \( 1 + 31.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 68.8iT - 9.40e3T^{2} \) |
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
−9.569685630235330344459354057280, −8.698901076310831406035431615101, −8.111808880854067137531073282232, −7.33178708081782557757782440329, −5.96601979470461822922464154317, −5.30162566783326354860589132622, −4.05395521316709410980121969433, −3.03839985979313990361050405652, −2.08697526359971213669698900958, −0.19404651800842916816896183676,
1.34893278771388993003728161632, 2.19976585210789011193959103439, 3.43796523018642802539564882361, 4.59666947976263970619896423169, 6.18628318430356652291042196585, 6.65298688355886171604129606076, 7.58139079304966892415009400047, 8.150478790881998655357456180112, 9.242557696594063766746583868810, 9.484045981253342590121644647572