| L(s) = 1 | + (−1.41 + 0.0614i)2-s + (1.84 + 1.15i)3-s + (1.99 − 0.173i)4-s + (1.46 − 1.17i)5-s + (−2.67 − 1.52i)6-s + (−1.73 + 0.396i)7-s + (−2.80 + 0.367i)8-s + (0.751 + 1.55i)9-s + (−2.00 + 1.74i)10-s + (3.52 − 1.23i)11-s + (3.87 + 1.98i)12-s + (−2.01 + 4.18i)13-s + (2.42 − 0.666i)14-s + (4.05 − 0.457i)15-s + (3.93 − 0.692i)16-s + (−3.39 − 3.39i)17-s + ⋯ |
| L(s) = 1 | + (−0.999 + 0.0434i)2-s + (1.06 + 0.668i)3-s + (0.996 − 0.0868i)4-s + (0.656 − 0.523i)5-s + (−1.09 − 0.621i)6-s + (−0.656 + 0.149i)7-s + (−0.991 + 0.130i)8-s + (0.250 + 0.519i)9-s + (−0.632 + 0.551i)10-s + (1.06 − 0.371i)11-s + (1.11 + 0.573i)12-s + (−0.559 + 1.16i)13-s + (0.649 − 0.178i)14-s + (1.04 − 0.118i)15-s + (0.984 − 0.173i)16-s + (−0.823 − 0.823i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 - 0.392i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.919 - 0.392i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.959698 + 0.196087i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.959698 + 0.196087i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.41 - 0.0614i)T \) |
| 29 | \( 1 + (5.33 - 0.698i)T \) |
| good | 3 | \( 1 + (-1.84 - 1.15i)T + (1.30 + 2.70i)T^{2} \) |
| 5 | \( 1 + (-1.46 + 1.17i)T + (1.11 - 4.87i)T^{2} \) |
| 7 | \( 1 + (1.73 - 0.396i)T + (6.30 - 3.03i)T^{2} \) |
| 11 | \( 1 + (-3.52 + 1.23i)T + (8.60 - 6.85i)T^{2} \) |
| 13 | \( 1 + (2.01 - 4.18i)T + (-8.10 - 10.1i)T^{2} \) |
| 17 | \( 1 + (3.39 + 3.39i)T + 17iT^{2} \) |
| 19 | \( 1 + (-2.19 - 3.49i)T + (-8.24 + 17.1i)T^{2} \) |
| 23 | \( 1 + (3.99 + 3.18i)T + (5.11 + 22.4i)T^{2} \) |
| 31 | \( 1 + (4.45 + 0.502i)T + (30.2 + 6.89i)T^{2} \) |
| 37 | \( 1 + (-2.08 + 5.97i)T + (-28.9 - 23.0i)T^{2} \) |
| 41 | \( 1 + (-5.95 + 5.95i)T - 41iT^{2} \) |
| 43 | \( 1 + (0.704 + 6.25i)T + (-41.9 + 9.56i)T^{2} \) |
| 47 | \( 1 + (-2.08 - 5.96i)T + (-36.7 + 29.3i)T^{2} \) |
| 53 | \( 1 + (0.431 + 0.541i)T + (-11.7 + 51.6i)T^{2} \) |
| 59 | \( 1 - 9.94iT - 59T^{2} \) |
| 61 | \( 1 + (-3.42 + 5.44i)T + (-26.4 - 54.9i)T^{2} \) |
| 67 | \( 1 + (4.28 - 2.06i)T + (41.7 - 52.3i)T^{2} \) |
| 71 | \( 1 + (-7.48 - 3.60i)T + (44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (-1.32 - 11.7i)T + (-71.1 + 16.2i)T^{2} \) |
| 79 | \( 1 + (0.994 - 2.84i)T + (-61.7 - 49.2i)T^{2} \) |
| 83 | \( 1 + (5.52 + 1.26i)T + (74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (-1.19 + 10.6i)T + (-86.7 - 19.8i)T^{2} \) |
| 97 | \( 1 + (-7.24 - 11.5i)T + (-42.0 + 87.3i)T^{2} \) |
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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
−14.00199013020364190055793638264, −12.47913447380742474513454724632, −11.35171225388220032673751203872, −9.839697790433871121139497943529, −9.251737670616015448931637460137, −8.851303465623320859453492185202, −7.25278270468959288651904572708, −5.95266857523437173143335288364, −3.85981960497228202465883155127, −2.19899190959004004157788810228,
1.99066769121397771026492968419, 3.23002153538788956030872361595, 6.16516808003354015192848313678, 7.12573721464350727595879796904, 8.092061743682059971862296385193, 9.301209033752562107010160383572, 9.941252427605872821401024176797, 11.20418381871519848924077392288, 12.60489838828430417764761856561, 13.43818094666065327931717200228
