| L(s) = 1 | + (−1.41 + 0.0275i)2-s + (1.72 − 0.179i)3-s + (1.99 − 0.0779i)4-s + (−0.211 − 1.60i)5-s + (−2.43 + 0.301i)6-s + (0.419 + 1.56i)7-s + (−2.82 + 0.165i)8-s + (2.93 − 0.619i)9-s + (0.343 + 2.26i)10-s + (4.18 − 3.21i)11-s + (3.42 − 0.493i)12-s + (−1.19 + 1.55i)13-s + (−0.636 − 2.20i)14-s + (−0.653 − 2.72i)15-s + (3.98 − 0.311i)16-s − 3.66·17-s + ⋯ |
| L(s) = 1 | + (−0.999 + 0.0194i)2-s + (0.994 − 0.103i)3-s + (0.999 − 0.0389i)4-s + (−0.0946 − 0.718i)5-s + (−0.992 + 0.123i)6-s + (0.158 + 0.592i)7-s + (−0.998 + 0.0584i)8-s + (0.978 − 0.206i)9-s + (0.108 + 0.716i)10-s + (1.26 − 0.969i)11-s + (0.989 − 0.142i)12-s + (−0.331 + 0.432i)13-s + (−0.170 − 0.588i)14-s + (−0.168 − 0.704i)15-s + (0.996 − 0.0778i)16-s − 0.890·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.896 + 0.442i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.896 + 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.18367 - 0.275917i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.18367 - 0.275917i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.41 - 0.0275i)T \) |
| 3 | \( 1 + (-1.72 + 0.179i)T \) |
| good | 5 | \( 1 + (0.211 + 1.60i)T + (-4.82 + 1.29i)T^{2} \) |
| 7 | \( 1 + (-0.419 - 1.56i)T + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (-4.18 + 3.21i)T + (2.84 - 10.6i)T^{2} \) |
| 13 | \( 1 + (1.19 - 1.55i)T + (-3.36 - 12.5i)T^{2} \) |
| 17 | \( 1 + 3.66T + 17T^{2} \) |
| 19 | \( 1 + (-0.398 + 0.165i)T + (13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (3.12 + 0.838i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (-6.15 - 0.809i)T + (28.0 + 7.50i)T^{2} \) |
| 31 | \( 1 + (1.59 - 0.919i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.24 + 3.00i)T + (-26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (1.46 - 5.45i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (3.53 + 4.60i)T + (-11.1 + 41.5i)T^{2} \) |
| 47 | \( 1 + (-2.19 - 1.26i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.87 - 11.7i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (9.75 - 1.28i)T + (56.9 - 15.2i)T^{2} \) |
| 61 | \( 1 + (1.45 - 11.0i)T + (-58.9 - 15.7i)T^{2} \) |
| 67 | \( 1 + (0.643 - 0.838i)T + (-17.3 - 64.7i)T^{2} \) |
| 71 | \( 1 + (11.7 + 11.7i)T + 71iT^{2} \) |
| 73 | \( 1 + (9.39 - 9.39i)T - 73iT^{2} \) |
| 79 | \( 1 + (1.60 - 2.78i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.65 - 0.218i)T + (80.1 + 21.4i)T^{2} \) |
| 89 | \( 1 + (8.98 - 8.98i)T - 89iT^{2} \) |
| 97 | \( 1 + (-5.58 + 9.68i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.81641965528494270702158370914, −10.62479404513554467420561088585, −9.381996258435949420986942812137, −8.839199716184361354640997450689, −8.355315713190210875473877045805, −7.09589614526158959537698649049, −6.10212470847719290577418230875, −4.32693186323124102725290348129, −2.81344112659558743764200228733, −1.40103095255847331843699572514,
1.75811124413931830686470823485, 3.09315129978976368199040511897, 4.36502277732857795166932428271, 6.57989007607975853299774307708, 7.16463974306742247604681560899, 8.080059539595038402313884840275, 9.102635196811198794846966079456, 9.912732668170763952229643693077, 10.59638223414098888454864531277, 11.66969743264600461558836068725
