L(s) = 1 | + (−0.866 − 0.5i)3-s + (1.29 + 2.23i)5-s + (−1.56 − 2.13i)7-s + (0.499 + 0.866i)9-s + (0.195 − 0.339i)11-s + 1.67·13-s − 2.58i·15-s + (0.678 + 0.391i)17-s + (4.19 − 2.42i)19-s + (0.292 + 2.62i)21-s + (−4.47 + 2.58i)23-s + (−0.839 + 1.45i)25-s − 0.999i·27-s + 1.79i·29-s + (1.17 − 2.03i)31-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.288i)3-s + (0.577 + 1.00i)5-s + (−0.592 − 0.805i)7-s + (0.166 + 0.288i)9-s + (0.0590 − 0.102i)11-s + 0.465·13-s − 0.667i·15-s + (0.164 + 0.0949i)17-s + (0.962 − 0.555i)19-s + (0.0637 + 0.573i)21-s + (−0.933 + 0.538i)23-s + (−0.167 + 0.290i)25-s − 0.192i·27-s + 0.333i·29-s + (0.211 − 0.365i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0244i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0244i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.496042669\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.496042669\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.866 + 0.5i)T \) |
| 7 | \( 1 + (1.56 + 2.13i)T \) |
good | 5 | \( 1 + (-1.29 - 2.23i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.195 + 0.339i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 1.67T + 13T^{2} \) |
| 17 | \( 1 + (-0.678 - 0.391i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.19 + 2.42i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.47 - 2.58i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 1.79iT - 29T^{2} \) |
| 31 | \( 1 + (-1.17 + 2.03i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.83 + 2.79i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 4.24iT - 41T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 + (-6.59 - 11.4i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-8.23 - 4.75i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (0.132 + 0.0766i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.26 - 3.91i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.46 + 4.27i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 0.524iT - 71T^{2} \) |
| 73 | \( 1 + (10.2 + 5.92i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.51 - 1.45i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 12.3iT - 83T^{2} \) |
| 89 | \( 1 + (-10.8 + 6.27i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 12.0iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.781613183307796609938869939786, −8.980358055622018589775844034150, −7.49178554002357747935293106958, −7.31215527756723652292463720168, −6.08644093938842427165244805437, −5.92436444723082111129716010099, −4.43771092464900742541156403333, −3.43446172787719204894405300275, −2.42710276736710643460507528456, −0.934925670472034472936855608146,
0.945352440431839622578255212504, 2.30526008167726928227078869854, 3.62382154016853816220881652316, 4.66085719335791406252091143775, 5.65419278653124174176123026110, 5.91370780886601191654849283374, 7.03616982903958934516049510237, 8.254580388222958278850735737685, 8.887351744406822009900877359609, 9.722999897795528074581159818971