| L(s) = 1 | + 2.23i·5-s − 1.18·7-s + 4.19i·11-s − 14.3·13-s − 18.7i·17-s + 33.8·19-s + 13.4i·23-s − 5.00·25-s + 35.3i·29-s − 9.95·31-s − 2.65i·35-s − 19.3·37-s + 64.6i·41-s − 41.5·43-s − 67.2i·47-s + ⋯ |
| L(s) = 1 | + 0.447i·5-s − 0.169·7-s + 0.381i·11-s − 1.10·13-s − 1.10i·17-s + 1.78·19-s + 0.586i·23-s − 0.200·25-s + 1.21i·29-s − 0.321·31-s − 0.0759i·35-s − 0.521·37-s + 1.57i·41-s − 0.966·43-s − 1.43i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2930201801\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2930201801\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
| good | 7 | \( 1 + 1.18T + 49T^{2} \) |
| 11 | \( 1 - 4.19iT - 121T^{2} \) |
| 13 | \( 1 + 14.3T + 169T^{2} \) |
| 17 | \( 1 + 18.7iT - 289T^{2} \) |
| 19 | \( 1 - 33.8T + 361T^{2} \) |
| 23 | \( 1 - 13.4iT - 529T^{2} \) |
| 29 | \( 1 - 35.3iT - 841T^{2} \) |
| 31 | \( 1 + 9.95T + 961T^{2} \) |
| 37 | \( 1 + 19.3T + 1.36e3T^{2} \) |
| 41 | \( 1 - 64.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 41.5T + 1.84e3T^{2} \) |
| 47 | \( 1 + 67.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 30.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 4.23iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 87.6T + 3.72e3T^{2} \) |
| 67 | \( 1 + 35.8T + 4.48e3T^{2} \) |
| 71 | \( 1 + 24.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 11.9T + 5.32e3T^{2} \) |
| 79 | \( 1 + 38.1T + 6.24e3T^{2} \) |
| 83 | \( 1 - 24.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 44.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 63.9T + 9.40e3T^{2} \) |
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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
−9.727088861590147919393791788103, −8.987960336157026369706047076773, −7.75451691930268600469843089225, −7.26555413065307701885628726711, −6.58324696405080353607862390930, −5.28831030183901105623235844326, −4.89616791430395970465931120116, −3.44667784082478963633791835474, −2.81246929104418143444466580505, −1.50618695073392450499621682358,
0.07809951492938472755739864091, 1.42226901505668344987003324538, 2.67265553290689237013548838806, 3.68977902510047096101014433848, 4.69456928129637594225876666923, 5.50699103202533101771987694552, 6.29938240472646230612847302876, 7.36550641335592307420248155167, 7.953166434863801783008952441088, 8.871330108763505845832483824516
