L(s) = 1 | + (0.633 − 2.36i)3-s + (2.23 − 0.133i)5-s + (−2 − 2i)7-s + (−2.59 − 1.50i)9-s + 11-s + (2.36 − 0.633i)13-s + (1.09 − 5.36i)15-s + (−0.366 + 1.36i)17-s + (−4.33 − 0.5i)19-s + (−6 + 3.46i)21-s + (4.96 − 0.598i)25-s + (−4.33 + 7.5i)29-s + 1.73i·31-s + (0.633 − 2.36i)33-s + (−4.73 − 4.19i)35-s + ⋯ |
L(s) = 1 | + (0.366 − 1.36i)3-s + (0.998 − 0.0599i)5-s + (−0.755 − 0.755i)7-s + (−0.866 − 0.500i)9-s + 0.301·11-s + (0.656 − 0.175i)13-s + (0.283 − 1.38i)15-s + (−0.0887 + 0.331i)17-s + (−0.993 − 0.114i)19-s + (−1.30 + 0.755i)21-s + (0.992 − 0.119i)25-s + (−0.804 + 1.39i)29-s + 0.311i·31-s + (0.110 − 0.411i)33-s + (−0.799 − 0.709i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.122 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.122 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.09338 - 1.23678i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.09338 - 1.23678i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2.23 + 0.133i)T \) |
| 19 | \( 1 + (4.33 + 0.5i)T \) |
good | 3 | \( 1 + (-0.633 + 2.36i)T + (-2.59 - 1.5i)T^{2} \) |
| 7 | \( 1 + (2 + 2i)T + 7iT^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 + (-2.36 + 0.633i)T + (11.2 - 6.5i)T^{2} \) |
| 17 | \( 1 + (0.366 - 1.36i)T + (-14.7 - 8.5i)T^{2} \) |
| 23 | \( 1 + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (4.33 - 7.5i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 1.73iT - 31T^{2} \) |
| 37 | \( 1 + (-3.46 + 3.46i)T - 37iT^{2} \) |
| 41 | \( 1 + (-3 + 1.73i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (5.46 + 1.46i)T + (37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (-4.09 + 1.09i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-7.09 + 1.90i)T + (45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (-0.866 - 1.5i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (6.5 - 11.2i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.16 - 11.8i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (-10.5 + 6.06i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-13.6 - 3.66i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-6.06 - 10.5i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (4 - 4i)T - 83iT^{2} \) |
| 89 | \( 1 + (7.79 - 13.5i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (18.9 + 5.07i)T + (84.0 + 48.5i)T^{2} \) |
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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
−11.02098739701082813591149014588, −10.22727733151913851640552101543, −9.122003384373383866096277086521, −8.328475992391639144265035343192, −7.04867479130688639665981883293, −6.61638033894151443048184960652, −5.60995023172593651087637825041, −3.82544727108315522112133042417, −2.40370342196760048394767382510, −1.18283700329449551835460499333,
2.31845141996152509949966139811, 3.50261109951135930513408051197, 4.62251144387401078142235911267, 5.82207365561848000148719741158, 6.50768420863319499381606035000, 8.279291753172397324165713384788, 9.345199422948549280028150008733, 9.485279929628718840464133187197, 10.47816793487332979644585187356, 11.32434771962609315668066954158