| L(s) = 1 | + (1.34 + 0.890i)2-s + (0.241 + 0.564i)4-s + (0.0720 − 0.0991i)5-s + (0.477 − 1.27i)7-s + (0.400 − 2.20i)8-s + (0.185 − 0.0696i)10-s + (0.263 − 1.94i)11-s + (3.81 − 0.517i)13-s + (1.77 − 1.29i)14-s + (3.35 − 3.50i)16-s + (−0.0384 + 0.118i)17-s + (1.18 − 0.705i)19-s + (0.0733 + 0.0167i)20-s + (2.08 − 2.38i)22-s + (−1.85 + 2.32i)23-s + ⋯ |
| L(s) = 1 | + (0.954 + 0.629i)2-s + (0.120 + 0.282i)4-s + (0.0322 − 0.0443i)5-s + (0.180 − 0.480i)7-s + (0.141 − 0.779i)8-s + (0.0586 − 0.0220i)10-s + (0.0794 − 0.586i)11-s + (1.05 − 0.143i)13-s + (0.474 − 0.344i)14-s + (0.838 − 0.876i)16-s + (−0.00931 + 0.0286i)17-s + (0.271 − 0.161i)19-s + (0.0164 + 0.00374i)20-s + (0.445 − 0.509i)22-s + (−0.387 + 0.485i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0501i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0501i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.50582 - 0.0629134i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.50582 - 0.0629134i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 71 | \( 1 + (-2.72 - 7.97i)T \) |
| good | 2 | \( 1 + (-1.34 - 0.890i)T + (0.786 + 1.83i)T^{2} \) |
| 5 | \( 1 + (-0.0720 + 0.0991i)T + (-1.54 - 4.75i)T^{2} \) |
| 7 | \( 1 + (-0.477 + 1.27i)T + (-5.27 - 4.60i)T^{2} \) |
| 11 | \( 1 + (-0.263 + 1.94i)T + (-10.6 - 2.92i)T^{2} \) |
| 13 | \( 1 + (-3.81 + 0.517i)T + (12.5 - 3.45i)T^{2} \) |
| 17 | \( 1 + (0.0384 - 0.118i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-1.18 + 0.705i)T + (9.00 - 16.7i)T^{2} \) |
| 23 | \( 1 + (1.85 - 2.32i)T + (-5.11 - 22.4i)T^{2} \) |
| 29 | \( 1 + (-0.924 - 1.05i)T + (-3.89 + 28.7i)T^{2} \) |
| 31 | \( 1 + (-2.37 + 2.26i)T + (1.39 - 30.9i)T^{2} \) |
| 37 | \( 1 + (1.95 + 2.44i)T + (-8.23 + 36.0i)T^{2} \) |
| 41 | \( 1 + (8.04 - 3.87i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (0.171 + 3.81i)T + (-42.8 + 3.85i)T^{2} \) |
| 47 | \( 1 + (-1.54 - 0.139i)T + (46.2 + 8.39i)T^{2} \) |
| 53 | \( 1 + (3.44 - 8.06i)T + (-36.6 - 38.3i)T^{2} \) |
| 59 | \( 1 + (-0.00866 - 0.0160i)T + (-32.5 + 49.2i)T^{2} \) |
| 61 | \( 1 + (2.87 + 7.65i)T + (-45.9 + 40.1i)T^{2} \) |
| 67 | \( 1 + (-6.97 + 2.98i)T + (46.3 - 48.4i)T^{2} \) |
| 73 | \( 1 + (0.962 - 1.45i)T + (-28.6 - 67.1i)T^{2} \) |
| 79 | \( 1 + (7.79 + 1.41i)T + (73.9 + 27.7i)T^{2} \) |
| 83 | \( 1 + (9.57 - 5.15i)T + (45.7 - 69.2i)T^{2} \) |
| 89 | \( 1 + (-0.893 - 0.381i)T + (61.5 + 64.3i)T^{2} \) |
| 97 | \( 1 + (1.89 - 3.93i)T + (-60.4 - 75.8i)T^{2} \) |
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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
−10.67685880314474668189863942018, −9.723008393296303027569076183221, −8.719880410563944580043251964667, −7.70481069218714144554729383844, −6.79821989790255524368930521441, −5.94560594141660879549648139495, −5.18112781480997037881705725741, −4.08177884708884530191662737565, −3.28434111079876734908588665019, −1.19174216207221722028043298129,
1.77853346704820141568320115304, 2.91611912947115189179069381040, 3.99007088018928760119048101167, 4.85213531716359970597224401904, 5.82155801086976101574710328061, 6.81346675025695997634570458051, 8.184339978575763773413888767460, 8.693503683159621579455209182377, 9.987243559607312458351526406721, 10.77050292556306480861217607872
