| L(s) = 1 | + 1.41i·2-s − 1.73·5-s + (1 + 2.44i)7-s + 2.82i·8-s − 2.44i·10-s + 1.41i·11-s + 2.44i·13-s + (−3.46 + 1.41i)14-s − 4.00·16-s + 5.19·17-s − 7.34i·19-s − 2.00·22-s − 2.82i·23-s − 2.00·25-s − 3.46·26-s + ⋯ |
| L(s) = 1 | + 0.999i·2-s − 0.774·5-s + (0.377 + 0.925i)7-s + 0.999i·8-s − 0.774i·10-s + 0.426i·11-s + 0.679i·13-s + (−0.925 + 0.377i)14-s − 1.00·16-s + 1.26·17-s − 1.68i·19-s − 0.426·22-s − 0.589i·23-s − 0.400·25-s − 0.679·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.377 - 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.377 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.658358 + 0.979882i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.658358 + 0.979882i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + (-1 - 2.44i)T \) |
| good | 2 | \( 1 - 1.41iT - 2T^{2} \) |
| 5 | \( 1 + 1.73T + 5T^{2} \) |
| 11 | \( 1 - 1.41iT - 11T^{2} \) |
| 13 | \( 1 - 2.44iT - 13T^{2} \) |
| 17 | \( 1 - 5.19T + 17T^{2} \) |
| 19 | \( 1 + 7.34iT - 19T^{2} \) |
| 23 | \( 1 + 2.82iT - 23T^{2} \) |
| 29 | \( 1 + 7.07iT - 29T^{2} \) |
| 31 | \( 1 - 2.44iT - 31T^{2} \) |
| 37 | \( 1 - 5T + 37T^{2} \) |
| 41 | \( 1 - 8.66T + 41T^{2} \) |
| 43 | \( 1 - 5T + 43T^{2} \) |
| 47 | \( 1 + 8.66T + 47T^{2} \) |
| 53 | \( 1 + 11.3iT - 53T^{2} \) |
| 59 | \( 1 - 8.66T + 59T^{2} \) |
| 61 | \( 1 + 2.44iT - 61T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 - 14.1iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 13T + 79T^{2} \) |
| 83 | \( 1 - 1.73T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 + 17.1iT - 97T^{2} \) |
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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
−12.83217219130144172886084074141, −11.67120188013919176218549325931, −11.33486793844748845137857339627, −9.639994425197906743321190836037, −8.507092058131594793711533661069, −7.72969249655452372306215068052, −6.75682140315435819717536517767, −5.60233471199409282489903453001, −4.47567587387250422591413427531, −2.50285706865570168890423261532,
1.21390625503354902557172281617, 3.26072537580287945895711806105, 4.06322499893154788370364151799, 5.81826367315714878428283556666, 7.42781759225687197034043682863, 8.006987996081415846114091344901, 9.700248769112340950507436858839, 10.52010450562826378801543801606, 11.25401025099405174551874117609, 12.13141041694563760704111852812
