L(s) = 1 | − 1.08i·2-s + 2.82·4-s − 3.33·5-s − 1.22·7-s − 7.40i·8-s + 3.62i·10-s + 1.21i·11-s − 12.6i·13-s + 1.33i·14-s + 3.23·16-s + 0.815·17-s − 7.74·19-s − 9.40·20-s + 1.32·22-s − 33.8i·23-s + ⋯ |
L(s) = 1 | − 0.543i·2-s + 0.705·4-s − 0.667·5-s − 0.175·7-s − 0.926i·8-s + 0.362i·10-s + 0.110i·11-s − 0.973i·13-s + 0.0951i·14-s + 0.202·16-s + 0.0479·17-s − 0.407·19-s − 0.470·20-s + 0.0601·22-s − 1.46i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.738 + 0.674i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.738 + 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.379308560\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.379308560\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 59 | \( 1 + (43.5 - 39.7i)T \) |
good | 2 | \( 1 + 1.08iT - 4T^{2} \) |
| 5 | \( 1 + 3.33T + 25T^{2} \) |
| 7 | \( 1 + 1.22T + 49T^{2} \) |
| 11 | \( 1 - 1.21iT - 121T^{2} \) |
| 13 | \( 1 + 12.6iT - 169T^{2} \) |
| 17 | \( 1 - 0.815T + 289T^{2} \) |
| 19 | \( 1 + 7.74T + 361T^{2} \) |
| 23 | \( 1 + 33.8iT - 529T^{2} \) |
| 29 | \( 1 + 7.53T + 841T^{2} \) |
| 31 | \( 1 + 7.71iT - 961T^{2} \) |
| 37 | \( 1 + 16.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 19.0T + 1.68e3T^{2} \) |
| 43 | \( 1 + 48.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 4.79iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 48.7T + 2.80e3T^{2} \) |
| 61 | \( 1 + 87.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 42.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 20.2T + 5.04e3T^{2} \) |
| 73 | \( 1 + 45.6iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 62.5T + 6.24e3T^{2} \) |
| 83 | \( 1 - 22.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 55.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 123. iT - 9.40e3T^{2} \) |
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
−10.53681590930392973138298116615, −9.661291785680430489685028781207, −8.423907373120456259035219841003, −7.60662259591129370900989032043, −6.70751976352612715844364490033, −5.71048644968505533209479366368, −4.27239038957388554752056826025, −3.28554605781136475405007623299, −2.19440289673321067225672158185, −0.51194596940299107784568935125,
1.72851770721700972225655318418, 3.16048690758936775753374396241, 4.33813224998593591868018194337, 5.61139272784680390965027191176, 6.51322759683024515773583889126, 7.36310377508322621466231233526, 8.043886396422117366818615627999, 9.071512935273913897608041568376, 10.08753197920124802458057895977, 11.33155026915151019452702783371