L(s) = 1 | − 3.61i·2-s + 15.5·3-s + 50.9·4-s + 42.6·5-s − 56.2i·6-s + 392. i·7-s − 415. i·8-s + 243·9-s − 154. i·10-s + (1.22e3 + 529. i)11-s + 794.·12-s − 2.61e3i·13-s + 1.41e3·14-s + 665.·15-s + 1.76e3·16-s − 7.83e3i·17-s + ⋯ |
L(s) = 1 | − 0.451i·2-s + 0.577·3-s + 0.796·4-s + 0.341·5-s − 0.260i·6-s + 1.14i·7-s − 0.810i·8-s + 0.333·9-s − 0.154i·10-s + (0.917 + 0.397i)11-s + 0.459·12-s − 1.18i·13-s + 0.516·14-s + 0.197·15-s + 0.430·16-s − 1.59i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 + 0.397i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.917 + 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(2.46615 - 0.511689i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.46615 - 0.511689i\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 15.5T \) |
| 11 | \( 1 + (-1.22e3 - 529. i)T \) |
good | 2 | \( 1 + 3.61iT - 64T^{2} \) |
| 5 | \( 1 - 42.6T + 1.56e4T^{2} \) |
| 7 | \( 1 - 392. iT - 1.17e5T^{2} \) |
| 13 | \( 1 + 2.61e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + 7.83e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 9.64e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 + 1.96e3T + 1.48e8T^{2} \) |
| 29 | \( 1 - 2.88e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 3.11e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + 7.13e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + 7.18e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 1.03e5iT - 6.32e9T^{2} \) |
| 47 | \( 1 - 5.34e4T + 1.07e10T^{2} \) |
| 53 | \( 1 - 5.78e4T + 2.21e10T^{2} \) |
| 59 | \( 1 + 1.28e5T + 4.21e10T^{2} \) |
| 61 | \( 1 + 5.90e4iT - 5.15e10T^{2} \) |
| 67 | \( 1 + 2.08e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + 6.21e5T + 1.28e11T^{2} \) |
| 73 | \( 1 - 1.25e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 + 4.67e4iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 4.96e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 1.66e5T + 4.96e11T^{2} \) |
| 97 | \( 1 - 1.49e6T + 8.32e11T^{2} \) |
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
−15.31718994766065509387057522591, −14.26450158996688714235121170020, −12.60561683078507801553386577309, −11.81961218068049589512877987394, −10.22499024806828519095934271779, −9.067515931107252248993669306770, −7.38930539604197212911733987884, −5.73545363345003556240599716424, −3.22050055387751544524956502636, −1.80968805233278152121600546465,
1.77784599995074291590159499694, 3.94192123976492620428799955276, 6.31117442659694773109810928014, 7.36051569250276911630130841683, 8.886670762057183771284047383554, 10.46436290327495887316449545751, 11.67950438710992875384534334381, 13.47384023015481220187348264422, 14.33070821055581571378099056462, 15.45375337134383867379965987773