| L(s) = 1 | + (0.981 + 1.01i)2-s + (0.971 + 0.402i)3-s + (−0.0713 + 1.99i)4-s + (−1.22 + 0.622i)5-s + (0.544 + 1.38i)6-s + (0.0357 + 0.0583i)7-s + (−2.10 + 1.89i)8-s + (−1.33 − 1.33i)9-s + (−1.83 − 0.631i)10-s + (5.58 + 0.439i)11-s + (−0.873 + 1.91i)12-s + (−1.03 − 4.32i)13-s + (−0.0242 + 0.0937i)14-s + (−1.43 + 0.113i)15-s + (−3.98 − 0.285i)16-s + (1.66 − 1.42i)17-s + ⋯ |
| L(s) = 1 | + (0.694 + 0.719i)2-s + (0.560 + 0.232i)3-s + (−0.0356 + 0.999i)4-s + (−0.545 + 0.278i)5-s + (0.222 + 0.564i)6-s + (0.0135 + 0.0220i)7-s + (−0.743 + 0.668i)8-s + (−0.446 − 0.446i)9-s + (−0.579 − 0.199i)10-s + (1.68 + 0.132i)11-s + (−0.252 + 0.552i)12-s + (−0.287 − 1.19i)13-s + (−0.00648 + 0.0250i)14-s + (−0.370 + 0.0291i)15-s + (−0.997 − 0.0713i)16-s + (0.404 − 0.345i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.30118 + 1.09693i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.30118 + 1.09693i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.981 - 1.01i)T \) |
| 41 | \( 1 + (-4.25 - 4.78i)T \) |
| good | 3 | \( 1 + (-0.971 - 0.402i)T + (2.12 + 2.12i)T^{2} \) |
| 5 | \( 1 + (1.22 - 0.622i)T + (2.93 - 4.04i)T^{2} \) |
| 7 | \( 1 + (-0.0357 - 0.0583i)T + (-3.17 + 6.23i)T^{2} \) |
| 11 | \( 1 + (-5.58 - 0.439i)T + (10.8 + 1.72i)T^{2} \) |
| 13 | \( 1 + (1.03 + 4.32i)T + (-11.5 + 5.90i)T^{2} \) |
| 17 | \( 1 + (-1.66 + 1.42i)T + (2.65 - 16.7i)T^{2} \) |
| 19 | \( 1 + (-1.78 - 0.428i)T + (16.9 + 8.62i)T^{2} \) |
| 23 | \( 1 + (-0.143 + 0.104i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (4.48 + 3.83i)T + (4.53 + 28.6i)T^{2} \) |
| 31 | \( 1 + (-0.761 - 2.34i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (1.58 - 4.88i)T + (-29.9 - 21.7i)T^{2} \) |
| 43 | \( 1 + (11.3 + 1.80i)T + (40.8 + 13.2i)T^{2} \) |
| 47 | \( 1 + (-2.70 + 4.42i)T + (-21.3 - 41.8i)T^{2} \) |
| 53 | \( 1 + (-2.51 + 2.94i)T + (-8.29 - 52.3i)T^{2} \) |
| 59 | \( 1 + (-2.06 - 2.84i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (12.9 - 2.04i)T + (58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (-0.475 - 6.04i)T + (-66.1 + 10.4i)T^{2} \) |
| 71 | \( 1 + (0.555 - 7.06i)T + (-70.1 - 11.1i)T^{2} \) |
| 73 | \( 1 + (-2.30 + 2.30i)T - 73iT^{2} \) |
| 79 | \( 1 + (4.26 - 10.2i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + 0.680iT - 83T^{2} \) |
| 89 | \( 1 + (3.76 - 2.30i)T + (40.4 - 79.2i)T^{2} \) |
| 97 | \( 1 + (-0.411 - 5.22i)T + (-95.8 + 15.1i)T^{2} \) |
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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
−13.30824216206818032737138421198, −11.99247867903555916349726499546, −11.59086981042951563229333245503, −9.788757872904379018637284311687, −8.753905585076940304734082795259, −7.78633201558440428465300787047, −6.73333296375177608909157554897, −5.51925782605107138797753804118, −3.95229261317788530799638386621, −3.14335141122826031478738817220,
1.78205484773867604245252872873, 3.48843320209554666090079152523, 4.50038648115469187969852248234, 6.02784093133481425987696541984, 7.31701858412128358692626805697, 8.801056877905278498394755560841, 9.477821706178386506622122988234, 10.99539682994059144284548443113, 11.77480501681593422455915400158, 12.45688997735333313085669486175
