L(s) = 1 | − 2.82i·2-s − 7.80i·3-s − 8.00·4-s − 33.0·5-s − 22.0·6-s + 16.0·7-s + 22.6i·8-s + 20.1·9-s + 93.6i·10-s − 215.·11-s + 62.4i·12-s − 281. i·13-s − 45.4i·14-s + 258. i·15-s + 64.0·16-s + 226.·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.867i·3-s − 0.500·4-s − 1.32·5-s − 0.613·6-s + 0.328·7-s + 0.353i·8-s + 0.248·9-s + 0.936i·10-s − 1.78·11-s + 0.433i·12-s − 1.66i·13-s − 0.232i·14-s + 1.14i·15-s + 0.250·16-s + 0.784·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0387i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0387i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.0155537 + 0.802003i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0155537 + 0.802003i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 2.82iT \) |
| 19 | \( 1 + (-13.9 + 360. i)T \) |
good | 3 | \( 1 + 7.80iT - 81T^{2} \) |
| 5 | \( 1 + 33.0T + 625T^{2} \) |
| 7 | \( 1 - 16.0T + 2.40e3T^{2} \) |
| 11 | \( 1 + 215.T + 1.46e4T^{2} \) |
| 13 | \( 1 + 281. iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 226.T + 8.35e4T^{2} \) |
| 23 | \( 1 - 414.T + 2.79e5T^{2} \) |
| 29 | \( 1 - 606. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 478. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 104. iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 1.89e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 315.T + 3.41e6T^{2} \) |
| 47 | \( 1 + 474.T + 4.87e6T^{2} \) |
| 53 | \( 1 + 774. iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 567. iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 4.79e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 4.46e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 7.63e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 8.39e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 9.70e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 1.05e4T + 4.74e7T^{2} \) |
| 89 | \( 1 + 1.02e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 1.54e4iT - 8.85e7T^{2} \) |
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
−15.02903363063193344973352300937, −13.14636067384833494790865700666, −12.69364924558045831202186445849, −11.38555878630890379595118880875, −10.31979767975201805991687228674, −8.129153141515518737636400381076, −7.54111090346297570572808495039, −5.04864726349963879376606642930, −2.99676737039772214886529090026, −0.56051810683536469085704849293,
3.89393897250238198753432320968, 5.07252253047712024793266339879, 7.28164823474562927200422926353, 8.317171134728054257926125755522, 9.887664685539706176412332400175, 11.17891234321650548528556370268, 12.56538116149668367564998660141, 14.19320065165023623846737036370, 15.34603367814436935349406273368, 15.95466063782112133522198189894