| L(s) = 1 | + (−1.47 + 2.61i)3-s + 2.23i·5-s + 2.64·7-s + (−4.67 − 7.68i)9-s + 13.0i·11-s + 10.2·13-s + (−5.84 − 3.28i)15-s − 21.6i·17-s − 21.6·19-s + (−3.89 + 6.91i)21-s − 33.9i·23-s − 5.00·25-s + (26.9 − 0.921i)27-s − 23.2i·29-s + 7.46·31-s + ⋯ |
| L(s) = 1 | + (−0.490 + 0.871i)3-s + 0.447i·5-s + 0.377·7-s + (−0.519 − 0.854i)9-s + 1.19i·11-s + 0.789·13-s + (−0.389 − 0.219i)15-s − 1.27i·17-s − 1.13·19-s + (−0.185 + 0.329i)21-s − 1.47i·23-s − 0.200·25-s + (0.999 − 0.0341i)27-s − 0.802i·29-s + 0.240·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 + 0.490i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.871 + 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.186992688\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.186992688\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.47 - 2.61i)T \) |
| 5 | \( 1 - 2.23iT \) |
| 7 | \( 1 - 2.64T \) |
| good | 11 | \( 1 - 13.0iT - 121T^{2} \) |
| 13 | \( 1 - 10.2T + 169T^{2} \) |
| 17 | \( 1 + 21.6iT - 289T^{2} \) |
| 19 | \( 1 + 21.6T + 361T^{2} \) |
| 23 | \( 1 + 33.9iT - 529T^{2} \) |
| 29 | \( 1 + 23.2iT - 841T^{2} \) |
| 31 | \( 1 - 7.46T + 961T^{2} \) |
| 37 | \( 1 - 29.2T + 1.36e3T^{2} \) |
| 41 | \( 1 + 6.98iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 51.1T + 1.84e3T^{2} \) |
| 47 | \( 1 + 43.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 16.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 32.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 15.5T + 3.72e3T^{2} \) |
| 67 | \( 1 + 34.1T + 4.48e3T^{2} \) |
| 71 | \( 1 + 127. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 78.3T + 5.32e3T^{2} \) |
| 79 | \( 1 + 67.0T + 6.24e3T^{2} \) |
| 83 | \( 1 + 81.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 139. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 128.T + 9.40e3T^{2} \) |
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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
−9.144518472067128166735966333925, −8.501889542239520643789815247546, −7.42192893796450654309828669027, −6.56905435886441986482376613782, −5.90525167704789829119653192873, −4.64647415706252630771076860922, −4.43030471095038025782898336576, −3.16033903977203177735673150279, −2.08765733290548440061663927595, −0.38520972994304185432129592198,
1.05091247850880121636218474420, 1.80297759816831795980184889099, 3.22366503279914018708970744849, 4.26892688826377416073194856603, 5.42507298864417467864264559787, 5.98790563128861854808625205941, 6.67598845017482355288735752597, 7.81813814450504717192673679578, 8.365185523084044798030516420423, 8.883710867622033262564614121930
