| L(s) = 1 | + (1.23 − 1.57i)2-s + (−1.38 − 2.66i)3-s + (−0.934 − 3.88i)4-s + (1.82 − 0.665i)5-s + (−5.89 − 1.12i)6-s + (0.628 − 0.110i)7-s + (−7.26 − 3.34i)8-s + (−5.18 + 7.35i)9-s + (1.21 − 3.69i)10-s + (1.29 − 3.56i)11-s + (−9.06 + 7.85i)12-s + (1.00 + 0.843i)13-s + (0.604 − 1.12i)14-s + (−4.29 − 3.95i)15-s + (−14.2 + 7.26i)16-s + (6.93 − 12.0i)17-s + ⋯ |
| L(s) = 1 | + (0.619 − 0.785i)2-s + (−0.460 − 0.887i)3-s + (−0.233 − 0.972i)4-s + (0.365 − 0.133i)5-s + (−0.982 − 0.188i)6-s + (0.0898 − 0.0158i)7-s + (−0.908 − 0.418i)8-s + (−0.576 + 0.817i)9-s + (0.121 − 0.369i)10-s + (0.118 − 0.324i)11-s + (−0.755 + 0.654i)12-s + (0.0773 + 0.0648i)13-s + (0.0431 − 0.0803i)14-s + (−0.286 − 0.263i)15-s + (−0.890 + 0.454i)16-s + (0.407 − 0.706i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.776 + 0.629i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.776 + 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.523070 - 1.47542i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.523070 - 1.47542i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.23 + 1.57i)T \) |
| 3 | \( 1 + (1.38 + 2.66i)T \) |
| good | 5 | \( 1 + (-1.82 + 0.665i)T + (19.1 - 16.0i)T^{2} \) |
| 7 | \( 1 + (-0.628 + 0.110i)T + (46.0 - 16.7i)T^{2} \) |
| 11 | \( 1 + (-1.29 + 3.56i)T + (-92.6 - 77.7i)T^{2} \) |
| 13 | \( 1 + (-1.00 - 0.843i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (-6.93 + 12.0i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-11.8 + 6.84i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-37.6 - 6.63i)T + (497. + 180. i)T^{2} \) |
| 29 | \( 1 + (-33.6 + 28.2i)T + (146. - 828. i)T^{2} \) |
| 31 | \( 1 + (6.16 + 1.08i)T + (903. + 328. i)T^{2} \) |
| 37 | \( 1 + (16.9 - 29.4i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-17.2 - 14.5i)T + (291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (-10.4 + 28.5i)T + (-1.41e3 - 1.18e3i)T^{2} \) |
| 47 | \( 1 + (52.0 - 9.18i)T + (2.07e3 - 755. i)T^{2} \) |
| 53 | \( 1 - 69.0T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-30.6 - 84.2i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-0.993 - 5.63i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (78.8 - 93.9i)T + (-779. - 4.42e3i)T^{2} \) |
| 71 | \( 1 + (88.3 + 51.0i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-33.2 - 57.5i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (66.1 + 78.8i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (3.87 + 4.61i)T + (-1.19e3 + 6.78e3i)T^{2} \) |
| 89 | \( 1 + (36.5 + 63.2i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-123. - 45.0i)T + (7.20e3 + 6.04e3i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.20592593655156390683877275138, −11.89888597393373198334065732390, −11.38473870771906522247538993131, −10.12738528891824064967183827384, −8.876441639525040719224398322116, −7.21556288753328155192874026485, −5.94205058547637334207324376680, −4.92969458757745523997517219155, −2.87950839743441530887476408353, −1.14388525540979582879873906976,
3.32700828714528020812600899052, 4.72044781074918598712121786941, 5.74559250882954806685296893147, 6.87628324571969449921339978958, 8.411289251325048100672372742172, 9.547395156205067864952742214817, 10.73434421868165863722758210559, 11.93833427835085822944069234392, 12.87939042616358347779211522676, 14.23367402329941581141707485574
