| L(s) = 1 | + (−1.40 − 0.129i)2-s + (1.33 − 1.10i)3-s + (1.96 + 0.363i)4-s + (0.297 + 0.0525i)5-s + (−2.02 + 1.38i)6-s + (0.312 + 0.371i)7-s + (−2.72 − 0.766i)8-s + (0.560 − 2.94i)9-s + (−0.412 − 0.112i)10-s + (0.00722 + 0.0410i)11-s + (3.02 − 1.68i)12-s + (4.05 − 1.47i)13-s + (−0.391 − 0.564i)14-s + (0.455 − 0.258i)15-s + (3.73 + 1.43i)16-s + (−2.14 − 1.23i)17-s + ⋯ |
| L(s) = 1 | + (−0.995 − 0.0913i)2-s + (0.770 − 0.637i)3-s + (0.983 + 0.181i)4-s + (0.133 + 0.0234i)5-s + (−0.825 + 0.564i)6-s + (0.117 + 0.140i)7-s + (−0.962 − 0.270i)8-s + (0.186 − 0.982i)9-s + (−0.130 − 0.0355i)10-s + (0.00217 + 0.0123i)11-s + (0.873 − 0.486i)12-s + (1.12 − 0.409i)13-s + (−0.104 − 0.150i)14-s + (0.117 − 0.0668i)15-s + (0.933 + 0.357i)16-s + (−0.520 − 0.300i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.784 + 0.619i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.784 + 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.834926 - 0.289820i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.834926 - 0.289820i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.40 + 0.129i)T \) |
| 3 | \( 1 + (-1.33 + 1.10i)T \) |
| good | 5 | \( 1 + (-0.297 - 0.0525i)T + (4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (-0.312 - 0.371i)T + (-1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (-0.00722 - 0.0410i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (-4.05 + 1.47i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (2.14 + 1.23i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (6.55 - 3.78i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.03 - 4.22i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (2.40 - 6.60i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (2.30 - 2.75i)T + (-5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (3.36 - 5.81i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.16 - 5.94i)T + (-31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-2.38 + 0.420i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-5.51 + 4.62i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + 9.90iT - 53T^{2} \) |
| 59 | \( 1 + (0.755 - 4.28i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (5.77 - 4.84i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (3.80 + 10.4i)T + (-51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (1.68 - 2.92i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.315 - 0.546i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.03 + 2.84i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (5.41 + 1.96i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (-9.00 + 5.19i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.578 + 3.28i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−13.46565763291206389634589989457, −12.57098761715600725720577138374, −11.37947338863558887627717063913, −10.28702967668265399858484597651, −8.978879004712179787390567544819, −8.369655790382608235008887307104, −7.18200613314458634959530961380, −6.07632664535769241762532317750, −3.41421420703452552872355039509, −1.74881089558505318338358892332,
2.27929197764220790175631587336, 4.13060876612320752754008711395, 6.10732455474398180930033949989, 7.50806780277998243886384357745, 8.732036037493762782199678684626, 9.217689709157709133815486832133, 10.65771625556809940745160374006, 11.10122016036460117854077081111, 12.87515072540625102475885450920, 13.99691867921782829700902169050
