L(s) = 1 | + (−0.177 + 0.0474i)2-s + (−0.650 − 1.60i)3-s + (−1.70 + 0.983i)4-s + (1.66 − 0.446i)5-s + (0.191 + 0.253i)6-s + (−0.707 − 0.707i)7-s + (0.514 − 0.514i)8-s + (−2.15 + 2.08i)9-s + (−0.273 + 0.158i)10-s + (0.724 + 2.70i)11-s + (2.68 + 2.09i)12-s + (1.29 + 3.36i)13-s + (0.158 + 0.0916i)14-s + (−1.79 − 2.38i)15-s + (1.89 − 3.29i)16-s + (3.00 − 5.21i)17-s + ⋯ |
L(s) = 1 | + (−0.125 + 0.0335i)2-s + (−0.375 − 0.926i)3-s + (−0.851 + 0.491i)4-s + (0.744 − 0.199i)5-s + (0.0780 + 0.103i)6-s + (−0.267 − 0.267i)7-s + (0.181 − 0.181i)8-s + (−0.718 + 0.695i)9-s + (−0.0865 + 0.0499i)10-s + (0.218 + 0.814i)11-s + (0.775 + 0.604i)12-s + (0.360 + 0.932i)13-s + (0.0424 + 0.0244i)14-s + (−0.464 − 0.615i)15-s + (0.474 − 0.822i)16-s + (0.729 − 1.26i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.854 + 0.518i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.854 + 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.11599 - 0.312152i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.11599 - 0.312152i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.650 + 1.60i)T \) |
| 7 | \( 1 + (0.707 + 0.707i)T \) |
| 13 | \( 1 + (-1.29 - 3.36i)T \) |
good | 2 | \( 1 + (0.177 - 0.0474i)T + (1.73 - i)T^{2} \) |
| 5 | \( 1 + (-1.66 + 0.446i)T + (4.33 - 2.5i)T^{2} \) |
| 11 | \( 1 + (-0.724 - 2.70i)T + (-9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (-3.00 + 5.21i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.241 - 0.899i)T + (-16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 - 5.41T + 23T^{2} \) |
| 29 | \( 1 + (-0.340 - 0.196i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.25 - 4.66i)T + (-26.8 + 15.5i)T^{2} \) |
| 37 | \( 1 + (-2.81 + 10.5i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (5.13 + 5.13i)T + 41iT^{2} \) |
| 43 | \( 1 + 10.1iT - 43T^{2} \) |
| 47 | \( 1 + (-12.1 - 3.25i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 - 1.96iT - 53T^{2} \) |
| 59 | \( 1 + (-3.58 - 0.961i)T + (51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 - 15.2T + 61T^{2} \) |
| 67 | \( 1 + (-6.17 + 6.17i)T - 67iT^{2} \) |
| 71 | \( 1 + (-0.488 + 0.130i)T + (61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (4.86 + 4.86i)T + 73iT^{2} \) |
| 79 | \( 1 + (-5.62 - 9.73i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (2.96 - 11.0i)T + (-71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + (14.3 + 3.84i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-9.08 + 9.08i)T - 97iT^{2} \) |
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
−9.946915608708860016564921910098, −9.245791011593006703438560924899, −8.590485554232565544613867994416, −7.22490016018113809756204236250, −7.10548368055614422596225263586, −5.67917348829213235330678207852, −4.98237794378488274056442215689, −3.75327237124884945436739282775, −2.29837381961440836610003131317, −0.924567461040402433998489359557,
0.984561362568658292275171665834, 2.98675025371467500459264623129, 3.96227215464788312736991012492, 5.12720677576767281100180793924, 5.83809130285909410479087722099, 6.32679587004677952117196886714, 8.211620591541174219952080310241, 8.719022612511536451009867472410, 9.801161976071019339871227204924, 10.04087516582181395140483850591