L(s) = 1 | + (−0.305 + 1.97i)2-s + 1.73·3-s + (−3.81 − 1.20i)4-s + (−0.529 + 3.42i)6-s + 0.329·7-s + (3.55 − 7.16i)8-s + 2.99·9-s + 20.4i·11-s + (−6.60 − 2.09i)12-s − 0.416i·13-s + (−0.100 + 0.652i)14-s + (13.0 + 9.21i)16-s + 18.5i·17-s + (−0.917 + 5.92i)18-s + 12.4i·19-s + ⋯ |
L(s) = 1 | + (−0.152 + 0.988i)2-s + 0.577·3-s + (−0.953 − 0.302i)4-s + (−0.0882 + 0.570i)6-s + 0.0471·7-s + (0.444 − 0.895i)8-s + 0.333·9-s + 1.86i·11-s + (−0.550 − 0.174i)12-s − 0.0320i·13-s + (−0.00720 + 0.0465i)14-s + (0.817 + 0.575i)16-s + 1.09i·17-s + (−0.0509 + 0.329i)18-s + 0.655i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.717 - 0.696i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.717 - 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.549169 + 1.35421i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.549169 + 1.35421i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.305 - 1.97i)T \) |
| 3 | \( 1 - 1.73T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 0.329T + 49T^{2} \) |
| 11 | \( 1 - 20.4iT - 121T^{2} \) |
| 13 | \( 1 + 0.416iT - 169T^{2} \) |
| 17 | \( 1 - 18.5iT - 289T^{2} \) |
| 19 | \( 1 - 12.4iT - 361T^{2} \) |
| 23 | \( 1 + 23.2T + 529T^{2} \) |
| 29 | \( 1 - 23.9T + 841T^{2} \) |
| 31 | \( 1 - 42.0iT - 961T^{2} \) |
| 37 | \( 1 + 50.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 46.7T + 1.68e3T^{2} \) |
| 43 | \( 1 + 55.5T + 1.84e3T^{2} \) |
| 47 | \( 1 - 81.7T + 2.20e3T^{2} \) |
| 53 | \( 1 + 29.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 24.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 74.8T + 3.72e3T^{2} \) |
| 67 | \( 1 - 72.8T + 4.48e3T^{2} \) |
| 71 | \( 1 + 39.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 46.5iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 101. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 5.88T + 6.88e3T^{2} \) |
| 89 | \( 1 - 61.0T + 7.92e3T^{2} \) |
| 97 | \( 1 + 95.5iT - 9.40e3T^{2} \) |
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
−12.34246242005556468923097304438, −10.44990725197283026570739822664, −9.874265349641358540812912378547, −8.870464841469313988281714255951, −7.939703846057187224231264137684, −7.18263776984898799710724534705, −6.13106971521525050414096337724, −4.79595816117410546408955742829, −3.85644715821131735858740175444, −1.81974822531947962653226259124,
0.73344348095053977453551050497, 2.53944535675987756744664578954, 3.46935236054594079043459510400, 4.72774144955877384885740337525, 6.08569901344175527844984269442, 7.71320256682818825977828943426, 8.529280168056121421336207861560, 9.301142165431993674076999949180, 10.25022630881969290470855914935, 11.28025149074382150630749095225