| L(s) = 1 | + i·2-s + (−0.721 − 1.57i)3-s − 4-s + (1.57 − 0.721i)6-s + (−1.31 + 2.29i)7-s − i·8-s + (−1.95 + 2.27i)9-s − 3.91i·11-s + (0.721 + 1.57i)12-s + 4.99i·13-s + (−2.29 − 1.31i)14-s + 16-s + 3.54·17-s + (−2.27 − 1.95i)18-s − 3.14i·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + (−0.416 − 0.909i)3-s − 0.5·4-s + (0.642 − 0.294i)6-s + (−0.496 + 0.867i)7-s − 0.353i·8-s + (−0.652 + 0.757i)9-s − 1.18i·11-s + (0.208 + 0.454i)12-s + 1.38i·13-s + (−0.613 − 0.351i)14-s + 0.250·16-s + 0.860·17-s + (−0.535 − 0.461i)18-s − 0.722i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0899 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0899 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6393145263\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6393145263\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + (0.721 + 1.57i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1.31 - 2.29i)T \) |
| good | 11 | \( 1 + 3.91iT - 11T^{2} \) |
| 13 | \( 1 - 4.99iT - 13T^{2} \) |
| 17 | \( 1 - 3.54T + 17T^{2} \) |
| 19 | \( 1 + 3.14iT - 19T^{2} \) |
| 23 | \( 1 + 7.54iT - 23T^{2} \) |
| 29 | \( 1 + 7.54iT - 29T^{2} \) |
| 31 | \( 1 + 4.19iT - 31T^{2} \) |
| 37 | \( 1 + 10.4T + 37T^{2} \) |
| 41 | \( 1 + 9.32T + 41T^{2} \) |
| 43 | \( 1 + 2.91T + 43T^{2} \) |
| 47 | \( 1 + 8.00T + 47T^{2} \) |
| 53 | \( 1 - 0.288iT - 53T^{2} \) |
| 59 | \( 1 - 5.89T + 59T^{2} \) |
| 61 | \( 1 + 2.48iT - 61T^{2} \) |
| 67 | \( 1 - 0.545T + 67T^{2} \) |
| 71 | \( 1 + 5.37iT - 71T^{2} \) |
| 73 | \( 1 + 4.85iT - 73T^{2} \) |
| 79 | \( 1 - 0.742T + 79T^{2} \) |
| 83 | \( 1 - 4.45T + 83T^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 + 0.524iT - 97T^{2} \) |
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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
−9.436532464307206658768004995540, −8.595250201391146561405826523941, −8.124173830747916729141474796253, −6.86164223992845623724128355250, −6.43416939953119687213207811054, −5.68415124167787086399133003714, −4.77361322643616952773953042359, −3.30930975316153323217875783456, −2.06829389242040314291249005723, −0.31753769077783261438014609348,
1.38924976510596922884669902560, 3.35971592558128000491386456495, 3.56137376964583380254074398501, 4.96903305048187364115507317282, 5.42755859379224433806433804616, 6.77789567177742632584997640483, 7.69035950343014821531485196653, 8.732132119220194522333556258798, 9.829417270946106511454061581481, 10.12910443499150900951828097992
