L(s) = 1 | − i·2-s + (−0.866 − 1.5i)3-s + 4-s + (−2.23 − 0.133i)5-s + (−1.5 + 0.866i)6-s + (−1.73 + 2i)7-s − 3i·8-s + (−1.5 + 2.59i)9-s + (−0.133 + 2.23i)10-s + (−3 − 5.19i)11-s + (−0.866 − 1.5i)12-s + (−3.46 + 2i)13-s + (2 + 1.73i)14-s + (1.73 + 3.46i)15-s − 16-s + (1.73 + i)17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (−0.499 − 0.866i)3-s + 0.5·4-s + (−0.998 − 0.0599i)5-s + (−0.612 + 0.353i)6-s + (−0.654 + 0.755i)7-s − 1.06i·8-s + (−0.5 + 0.866i)9-s + (−0.0423 + 0.705i)10-s + (−0.904 − 1.56i)11-s + (−0.249 − 0.433i)12-s + (−0.960 + 0.554i)13-s + (0.534 + 0.462i)14-s + (0.447 + 0.894i)15-s − 0.250·16-s + (0.420 + 0.242i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.883 - 0.468i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.883 - 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.123871 + 0.497572i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.123871 + 0.497572i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.866 + 1.5i)T \) |
| 5 | \( 1 + (2.23 + 0.133i)T \) |
| 7 | \( 1 + (1.73 - 2i)T \) |
good | 2 | \( 1 + iT - 2T^{2} \) |
| 11 | \( 1 + (3 + 5.19i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (3.46 - 2i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.73 - i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3 + 5.19i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.59 - 1.5i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1 - 1.73i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + (-6.92 + 4i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1 + 1.73i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.866 + 0.5i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 3iT - 47T^{2} \) |
| 53 | \( 1 + (10.3 + 6i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 7T + 61T^{2} \) |
| 67 | \( 1 + 7iT - 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 14T + 79T^{2} \) |
| 83 | \( 1 + (3.46 + 2i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (1.5 + 2.59i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.73 - i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.21733018733375730561097664122, −10.76210384556444385208124954613, −9.259941338152964556907249640285, −8.152309216388511979300039116679, −7.19330332384531695378771713515, −6.32566613803766210978412408224, −5.16772431360540474487607417530, −3.33336188981734894675419931831, −2.40383248464456408378155162931, −0.36062599657678425611783671956,
2.90851501749655848745796840625, 4.29632932896992717040354542330, 5.18890549455590836968474999796, 6.47784088149282347659265848763, 7.41742753396503173266614618721, 7.995865850317798147103130118861, 9.658357538816125916307232406853, 10.35479311746043754959145812859, 11.09409348312727556683372101890, 12.24479355221013431836131454166