L(s) = 1 | + i·2-s + (−1.55 − 0.758i)3-s − 4-s − 2.55·5-s + (0.758 − 1.55i)6-s + (0.900 + 2.48i)7-s − i·8-s + (1.84 + 2.36i)9-s − 2.55i·10-s − 6.10i·11-s + (1.55 + 0.758i)12-s − 4.03i·13-s + (−2.48 + 0.900i)14-s + (3.97 + 1.93i)15-s + 16-s − 1.51·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (−0.899 − 0.437i)3-s − 0.5·4-s − 1.14·5-s + (0.309 − 0.635i)6-s + (0.340 + 0.940i)7-s − 0.353i·8-s + (0.616 + 0.787i)9-s − 0.807i·10-s − 1.84i·11-s + (0.449 + 0.218i)12-s − 1.11i·13-s + (−0.664 + 0.240i)14-s + (1.02 + 0.500i)15-s + 0.250·16-s − 0.367·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.105 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.105 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.519806 + 0.467459i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.519806 + 0.467459i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.55 + 0.758i)T \) |
| 7 | \( 1 + (-0.900 - 2.48i)T \) |
| 23 | \( 1 - iT \) |
good | 5 | \( 1 + 2.55T + 5T^{2} \) |
| 11 | \( 1 + 6.10iT - 11T^{2} \) |
| 13 | \( 1 + 4.03iT - 13T^{2} \) |
| 17 | \( 1 + 1.51T + 17T^{2} \) |
| 19 | \( 1 - 4.48iT - 19T^{2} \) |
| 29 | \( 1 - 6.55iT - 29T^{2} \) |
| 31 | \( 1 - 4.67iT - 31T^{2} \) |
| 37 | \( 1 - 9.05T + 37T^{2} \) |
| 41 | \( 1 - 6.16T + 41T^{2} \) |
| 43 | \( 1 + 3.13T + 43T^{2} \) |
| 47 | \( 1 - 0.328T + 47T^{2} \) |
| 53 | \( 1 - 13.7iT - 53T^{2} \) |
| 59 | \( 1 - 13.6T + 59T^{2} \) |
| 61 | \( 1 - 3.68iT - 61T^{2} \) |
| 67 | \( 1 - 6.92T + 67T^{2} \) |
| 71 | \( 1 - 11.8iT - 71T^{2} \) |
| 73 | \( 1 + 6.01iT - 73T^{2} \) |
| 79 | \( 1 + 4.13T + 79T^{2} \) |
| 83 | \( 1 - 13.6T + 83T^{2} \) |
| 89 | \( 1 + 5.12T + 89T^{2} \) |
| 97 | \( 1 - 4.80iT - 97T^{2} \) |
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
−10.46955929881025282178579697432, −9.054502742916975124507936731450, −8.091318124370828472803225984315, −7.957588851183562315035295440427, −6.74971198483496122537190625769, −5.73088634653574198433585154577, −5.45780955417411223343879395619, −4.16006927294212102158549787830, −3.01682753581155754793098583647, −0.932464646482551237500848338171,
0.51087132162813640073986917940, 2.11940919223603153637529400897, 4.01610356631791582199058651321, 4.25907834854534131990068968387, 4.94853751225096128612498931919, 6.59335017277634404975192304205, 7.23054892443984035534391556427, 8.083626852640853642769182473213, 9.498139814448614070619526030378, 9.807685305627740804808416727588