| L(s) = 1 | + (1.41 + 0.0318i)2-s + (0.260 + 0.277i)3-s + (1.99 + 0.0901i)4-s + (−1.25 − 2.75i)5-s + (0.359 + 0.401i)6-s + (−2.56 + 0.660i)7-s + (2.82 + 0.191i)8-s + (0.186 − 2.84i)9-s + (−1.67 − 3.94i)10-s + (−1.27 − 2.04i)11-s + (0.495 + 0.578i)12-s + (−0.535 − 5.43i)13-s + (−3.64 + 0.851i)14-s + (0.441 − 1.06i)15-s + (3.98 + 0.360i)16-s + (−6.49 − 0.855i)17-s + ⋯ |
| L(s) = 1 | + (0.999 + 0.0225i)2-s + (0.150 + 0.160i)3-s + (0.998 + 0.0450i)4-s + (−0.559 − 1.23i)5-s + (0.146 + 0.163i)6-s + (−0.968 + 0.249i)7-s + (0.997 + 0.0675i)8-s + (0.0622 − 0.949i)9-s + (−0.531 − 1.24i)10-s + (−0.384 − 0.617i)11-s + (0.142 + 0.167i)12-s + (−0.148 − 1.50i)13-s + (−0.973 + 0.227i)14-s + (0.114 − 0.275i)15-s + (0.995 + 0.0900i)16-s + (−1.57 − 0.207i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.122 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.122 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.45038 - 1.63978i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.45038 - 1.63978i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.41 - 0.0318i)T \) |
| 7 | \( 1 + (2.56 - 0.660i)T \) |
| good | 3 | \( 1 + (-0.260 - 0.277i)T + (-0.196 + 2.99i)T^{2} \) |
| 5 | \( 1 + (1.25 + 2.75i)T + (-3.29 + 3.75i)T^{2} \) |
| 11 | \( 1 + (1.27 + 2.04i)T + (-4.86 + 9.86i)T^{2} \) |
| 13 | \( 1 + (0.535 + 5.43i)T + (-12.7 + 2.53i)T^{2} \) |
| 17 | \( 1 + (6.49 + 0.855i)T + (16.4 + 4.39i)T^{2} \) |
| 19 | \( 1 + (1.11 - 6.75i)T + (-17.9 - 6.10i)T^{2} \) |
| 23 | \( 1 + (-0.655 - 0.747i)T + (-3.00 + 22.8i)T^{2} \) |
| 29 | \( 1 + (-2.92 - 5.48i)T + (-16.1 + 24.1i)T^{2} \) |
| 31 | \( 1 + (-5.99 + 1.60i)T + (26.8 - 15.5i)T^{2} \) |
| 37 | \( 1 + (-3.31 + 8.81i)T + (-27.8 - 24.3i)T^{2} \) |
| 41 | \( 1 + (0.166 + 0.838i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (-4.71 + 1.43i)T + (35.7 - 23.8i)T^{2} \) |
| 47 | \( 1 + (-1.72 + 2.25i)T + (-12.1 - 45.3i)T^{2} \) |
| 53 | \( 1 + (-5.50 - 8.85i)T + (-23.4 + 47.5i)T^{2} \) |
| 59 | \( 1 + (-0.336 + 0.470i)T + (-18.9 - 55.8i)T^{2} \) |
| 61 | \( 1 + (-11.7 + 2.74i)T + (54.7 - 26.9i)T^{2} \) |
| 67 | \( 1 + (4.88 + 5.21i)T + (-4.38 + 66.8i)T^{2} \) |
| 71 | \( 1 + (-1.54 - 2.30i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (8.26 - 4.07i)T + (44.4 - 57.9i)T^{2} \) |
| 79 | \( 1 + (1.12 + 8.51i)T + (-76.3 + 20.4i)T^{2} \) |
| 83 | \( 1 + (7.39 - 9.01i)T + (-16.1 - 81.4i)T^{2} \) |
| 89 | \( 1 + (1.92 + 5.66i)T + (-70.6 + 54.1i)T^{2} \) |
| 97 | \( 1 + (0.0850 - 0.0850i)T - 97iT^{2} \) |
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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
−10.00124213356431559684836819881, −8.882357560769172854856462168518, −8.286361834517977981221102242840, −7.22073419353576210534672238022, −6.10198785635849580383953386429, −5.54260051170642357216398353709, −4.38920913778511056357668955757, −3.63117173234329882220749119075, −2.68340544147184790660420783638, −0.70681569087627080432695540924,
2.34945697400707041795846924601, 2.77015029685731250182640315709, 4.22264414785514850108896987394, 4.66298975162282010060753200585, 6.35553472789530187248595169263, 6.86064806293269876319175706969, 7.27002629530436379721521307819, 8.497691309256428496105809262215, 9.812274021139502639445158568833, 10.56011341020615416982228716051
