L(s) = 1 | − 4.54e9i·2-s + (5.46e15 − 1.01e15i)3-s + 5.31e19·4-s − 1.75e23i·5-s + (−4.61e24 − 2.48e25i)6-s − 8.19e27·7-s − 5.76e29i·8-s + (2.88e31 − 1.11e31i)9-s − 7.95e32·10-s − 2.37e34i·11-s + (2.90e35 − 5.40e34i)12-s + 7.13e36·13-s + 3.71e37i·14-s + (−1.78e38 − 9.58e38i)15-s + 1.30e39·16-s − 5.94e40i·17-s + ⋯ |
L(s) = 1 | − 0.528i·2-s + (0.983 − 0.182i)3-s + 0.720·4-s − 1.50i·5-s + (−0.0966 − 0.519i)6-s − 1.05·7-s − 0.909i·8-s + (0.933 − 0.359i)9-s − 0.795·10-s − 1.02i·11-s + (0.708 − 0.131i)12-s + 1.23·13-s + 0.560i·14-s + (−0.275 − 1.48i)15-s + 0.240·16-s − 1.47i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.182i)\, \overline{\Lambda}(67-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+33) \, L(s)\cr =\mathstrut & (-0.983 + 0.182i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{67}{2})\) |
\(\approx\) |
\(3.721082326\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.721082326\) |
\(L(34)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-5.46e15 + 1.01e15i)T \) |
good | 2 | \( 1 + 4.54e9iT - 7.37e19T^{2} \) |
| 5 | \( 1 + 1.75e23iT - 1.35e46T^{2} \) |
| 7 | \( 1 + 8.19e27T + 5.97e55T^{2} \) |
| 11 | \( 1 + 2.37e34iT - 5.39e68T^{2} \) |
| 13 | \( 1 - 7.13e36T + 3.31e73T^{2} \) |
| 17 | \( 1 + 5.94e40iT - 1.62e81T^{2} \) |
| 19 | \( 1 - 8.13e41T + 2.49e84T^{2} \) |
| 23 | \( 1 - 1.37e45iT - 7.48e89T^{2} \) |
| 29 | \( 1 - 9.76e47iT - 3.29e96T^{2} \) |
| 31 | \( 1 + 1.85e49T + 2.69e98T^{2} \) |
| 37 | \( 1 - 6.12e51T + 3.17e103T^{2} \) |
| 41 | \( 1 + 1.20e53iT - 2.77e106T^{2} \) |
| 43 | \( 1 + 2.91e53T + 6.44e107T^{2} \) |
| 47 | \( 1 - 1.66e54iT - 2.28e110T^{2} \) |
| 53 | \( 1 - 6.92e56iT - 6.34e113T^{2} \) |
| 59 | \( 1 - 2.60e58iT - 7.52e116T^{2} \) |
| 61 | \( 1 - 1.11e59T + 6.78e117T^{2} \) |
| 67 | \( 1 - 3.96e59T + 3.31e120T^{2} \) |
| 71 | \( 1 - 1.42e61iT - 1.52e122T^{2} \) |
| 73 | \( 1 + 2.22e61T + 9.53e122T^{2} \) |
| 79 | \( 1 + 3.86e62T + 1.75e125T^{2} \) |
| 83 | \( 1 + 9.18e62iT - 4.56e126T^{2} \) |
| 89 | \( 1 + 3.59e64iT - 4.56e128T^{2} \) |
| 97 | \( 1 + 5.76e65T + 1.33e131T^{2} \) |
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
−12.87059619014701935920057929851, −11.51278833989287835869317614138, −9.636393567183491207904689883284, −8.804834016833050100335662516488, −7.30578808293626948236214557942, −5.73745534206022690176910619036, −3.76248354541741762575500234510, −2.96463008317429504944007540461, −1.41468659267835547926094505554, −0.71400622360229106883414514458,
1.87755119236548036124540485425, 2.85353845441109223133671342196, 3.80356020921714812886520557754, 6.25588061921126236059699854627, 6.91852591160039405545271771914, 8.160593769973988229956671533135, 9.932561970947997023307737586624, 10.89761800429581062318530533043, 12.90485065760264586374074451874, 14.51974998773165908082091719816