| L(s) = 1 | + (−1.00 + 0.999i)2-s + (1.13 − 2.72i)3-s + (0.00192 − 1.99i)4-s + (−0.430 + 0.844i)5-s + (1.59 + 3.86i)6-s + (0.0257 + 0.107i)7-s + (1.99 + 2.00i)8-s + (−4.04 − 4.04i)9-s + (−0.413 − 1.27i)10-s + (3.23 − 3.78i)11-s + (−5.45 − 2.26i)12-s + (−2.01 − 3.28i)13-s + (−0.132 − 0.0814i)14-s + (1.81 + 2.12i)15-s + (−3.99 − 0.00768i)16-s + (0.108 + 1.38i)17-s + ⋯ |
| L(s) = 1 | + (−0.707 + 0.706i)2-s + (0.652 − 1.57i)3-s + (0.000960 − 0.999i)4-s + (−0.192 + 0.377i)5-s + (0.651 + 1.57i)6-s + (0.00972 + 0.0404i)7-s + (0.706 + 0.708i)8-s + (−1.34 − 1.34i)9-s + (−0.130 − 0.403i)10-s + (0.974 − 1.14i)11-s + (−1.57 − 0.654i)12-s + (−0.557 − 0.909i)13-s + (−0.0354 − 0.0217i)14-s + (0.469 + 0.549i)15-s + (−0.999 − 0.00192i)16-s + (0.0263 + 0.335i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.514 + 0.857i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.514 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.823141 - 0.465908i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.823141 - 0.465908i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.00 - 0.999i)T \) |
| 41 | \( 1 + (0.811 - 6.35i)T \) |
| good | 3 | \( 1 + (-1.13 + 2.72i)T + (-2.12 - 2.12i)T^{2} \) |
| 5 | \( 1 + (0.430 - 0.844i)T + (-2.93 - 4.04i)T^{2} \) |
| 7 | \( 1 + (-0.0257 - 0.107i)T + (-6.23 + 3.17i)T^{2} \) |
| 11 | \( 1 + (-3.23 + 3.78i)T + (-1.72 - 10.8i)T^{2} \) |
| 13 | \( 1 + (2.01 + 3.28i)T + (-5.90 + 11.5i)T^{2} \) |
| 17 | \( 1 + (-0.108 - 1.38i)T + (-16.7 + 2.65i)T^{2} \) |
| 19 | \( 1 + (3.34 + 2.05i)T + (8.62 + 16.9i)T^{2} \) |
| 23 | \( 1 + (-6.86 - 4.98i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (0.497 - 6.32i)T + (-28.6 - 4.53i)T^{2} \) |
| 31 | \( 1 + (0.564 - 1.73i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-0.756 - 2.32i)T + (-29.9 + 21.7i)T^{2} \) |
| 43 | \( 1 + (0.666 + 4.20i)T + (-40.8 + 13.2i)T^{2} \) |
| 47 | \( 1 + (-0.398 + 1.66i)T + (-41.8 - 21.3i)T^{2} \) |
| 53 | \( 1 + (-9.27 - 0.730i)T + (52.3 + 8.29i)T^{2} \) |
| 59 | \( 1 + (4.07 - 5.60i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (1.41 - 8.90i)T + (-58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (-10.1 + 8.66i)T + (10.4 - 66.1i)T^{2} \) |
| 71 | \( 1 + (6.45 + 5.51i)T + (11.1 + 70.1i)T^{2} \) |
| 73 | \( 1 + (6.92 - 6.92i)T - 73iT^{2} \) |
| 79 | \( 1 + (7.44 + 3.08i)T + (55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + 3.71iT - 83T^{2} \) |
| 89 | \( 1 + (15.9 - 3.82i)T + (79.2 - 40.4i)T^{2} \) |
| 97 | \( 1 + (-5.75 + 4.91i)T + (15.1 - 95.8i)T^{2} \) |
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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
−12.96685769092467010635895676735, −11.66631482156230061176856565811, −10.68795307000135903907505456538, −9.035665491254852518982115135806, −8.490572935692650692386282246323, −7.33044321097118058418979785173, −6.78924323291502510395650582415, −5.60777827680152304543227153853, −3.03403478523589672796085788967, −1.20616365159513252809483809667,
2.43577380765843940067250064143, 4.09650556836429666887112909211, 4.56553621919404678004158842223, 7.00726998528713589005332288114, 8.448742659065980522118583160014, 9.195603365681230079532496176784, 9.811994125678790018912643925209, 10.72830510740631105746904744507, 11.78554333133610257500614497780, 12.71309306757370252404633762217
