L(s) = 1 | + (0.568 − 0.328i)2-s + (−0.784 + 1.35i)4-s + (1.65 + 2.86i)5-s + (−2.58 − 0.568i)7-s + 2.34i·8-s + (1.88 + 1.08i)10-s + (2.02 + 1.17i)11-s − 1.58i·13-s + (−1.65 + 0.524i)14-s + (−0.799 − 1.38i)16-s + (0.568 − 0.984i)17-s + (3.85 − 2.22i)19-s − 5.19·20-s + 1.53·22-s + (8.13 − 4.69i)23-s + ⋯ |
L(s) = 1 | + (0.402 − 0.232i)2-s + (−0.392 + 0.679i)4-s + (0.740 + 1.28i)5-s + (−0.976 − 0.214i)7-s + 0.828i·8-s + (0.595 + 0.343i)10-s + (0.611 + 0.353i)11-s − 0.438i·13-s + (−0.442 + 0.140i)14-s + (−0.199 − 0.346i)16-s + (0.137 − 0.238i)17-s + (0.884 − 0.510i)19-s − 1.16·20-s + 0.328·22-s + (1.69 − 0.979i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.575 - 0.817i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.575 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.20488 + 0.625233i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.20488 + 0.625233i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (2.58 + 0.568i)T \) |
good | 2 | \( 1 + (-0.568 + 0.328i)T + (1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (-1.65 - 2.86i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.02 - 1.17i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 1.58iT - 13T^{2} \) |
| 17 | \( 1 + (-0.568 + 0.984i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.85 + 2.22i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-8.13 + 4.69i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 3.65iT - 29T^{2} \) |
| 31 | \( 1 + (6.33 + 3.65i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.58 - 4.47i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 9.64T + 41T^{2} \) |
| 43 | \( 1 + 2.16T + 43T^{2} \) |
| 47 | \( 1 + (2.79 + 4.83i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-8.70 - 5.02i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.08 + 1.88i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.46 + 2.00i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.38 + 9.32i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 6.39iT - 71T^{2} \) |
| 73 | \( 1 + (9.25 + 5.34i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.616 + 1.06i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 1.03T + 83T^{2} \) |
| 89 | \( 1 + (-3.73 - 6.46i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 13.5iT - 97T^{2} \) |
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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
−12.94274596005260301966354215995, −11.80952552274336056431449746708, −10.78580810677736569879285630831, −9.809356224229891766186131171008, −8.914367053921505245292591462751, −7.28124101071169915718286818067, −6.62026284753832336757874602066, −5.17166649416870479313682152561, −3.51499434626570847547450702740, −2.73381909199278755860885802027,
1.28646409251738375091327508019, 3.69830741621834877958108922721, 5.15352584390238513922356248525, 5.79172972445034783562184033518, 6.92960193812229129517934150946, 8.838735311133403304916159826006, 9.332793364366319304133101240835, 10.10583612909667517834746960573, 11.64926807033391773527589133498, 12.83587594454581943392825794107