| L(s) = 1 | + (0.827 − 1.14i)2-s + (0.881 − 0.471i)3-s + (−0.632 − 1.89i)4-s + (2.75 + 3.35i)5-s + (0.188 − 1.40i)6-s + (−1.44 + 0.963i)7-s + (−2.69 − 0.844i)8-s + (0.555 − 0.831i)9-s + (6.12 − 0.384i)10-s + (4.36 − 1.32i)11-s + (−1.45 − 1.37i)12-s + (−0.620 − 0.509i)13-s + (−0.0871 + 2.45i)14-s + (4.01 + 1.66i)15-s + (−3.20 + 2.39i)16-s + (2.30 − 0.956i)17-s + ⋯ |
| L(s) = 1 | + (0.584 − 0.811i)2-s + (0.509 − 0.272i)3-s + (−0.316 − 0.948i)4-s + (1.23 + 1.50i)5-s + (0.0769 − 0.572i)6-s + (−0.544 + 0.364i)7-s + (−0.954 − 0.298i)8-s + (0.185 − 0.277i)9-s + (1.93 − 0.121i)10-s + (1.31 − 0.399i)11-s + (−0.419 − 0.397i)12-s + (−0.172 − 0.141i)13-s + (−0.0233 + 0.654i)14-s + (1.03 + 0.428i)15-s + (−0.800 + 0.599i)16-s + (0.560 − 0.232i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.631 + 0.775i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.631 + 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.13190 - 1.01292i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.13190 - 1.01292i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.827 + 1.14i)T \) |
| 3 | \( 1 + (-0.881 + 0.471i)T \) |
| good | 5 | \( 1 + (-2.75 - 3.35i)T + (-0.975 + 4.90i)T^{2} \) |
| 7 | \( 1 + (1.44 - 0.963i)T + (2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (-4.36 + 1.32i)T + (9.14 - 6.11i)T^{2} \) |
| 13 | \( 1 + (0.620 + 0.509i)T + (2.53 + 12.7i)T^{2} \) |
| 17 | \( 1 + (-2.30 + 0.956i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (0.650 + 6.60i)T + (-18.6 + 3.70i)T^{2} \) |
| 23 | \( 1 + (8.19 - 1.63i)T + (21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (1.28 - 4.23i)T + (-24.1 - 16.1i)T^{2} \) |
| 31 | \( 1 + (-0.608 - 0.608i)T + 31iT^{2} \) |
| 37 | \( 1 + (5.62 + 0.553i)T + (36.2 + 7.21i)T^{2} \) |
| 41 | \( 1 + (0.680 + 3.42i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (-1.97 - 1.05i)T + (23.8 + 35.7i)T^{2} \) |
| 47 | \( 1 + (1.97 + 4.77i)T + (-33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (3.50 + 11.5i)T + (-44.0 + 29.4i)T^{2} \) |
| 59 | \( 1 + (0.265 - 0.218i)T + (11.5 - 57.8i)T^{2} \) |
| 61 | \( 1 + (-5.70 - 10.6i)T + (-33.8 + 50.7i)T^{2} \) |
| 67 | \( 1 + (-2.13 - 3.99i)T + (-37.2 + 55.7i)T^{2} \) |
| 71 | \( 1 + (1.73 + 2.59i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (3.87 + 2.58i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (1.12 - 2.72i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (3.76 - 0.370i)T + (81.4 - 16.1i)T^{2} \) |
| 89 | \( 1 + (-8.81 - 1.75i)T + (82.2 + 34.0i)T^{2} \) |
| 97 | \( 1 + (-0.442 - 0.442i)T + 97iT^{2} \) |
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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
−11.25422495181704319376680497822, −10.25992566173305260329383173944, −9.614645777980867990623617005355, −8.908857728976771633082533844814, −7.00878672754113519024063429281, −6.38900732553524143584335214147, −5.51255119311623678734486702087, −3.67179583447648556678122991788, −2.86358009227586454001273466622, −1.85209650398546405236607924791,
1.82760940291487278531100829488, 3.78834890896637205637160571599, 4.53258547843263906922569121397, 5.78856726868836919874296266172, 6.38258498626172312885922292287, 7.87308436713083914146483051445, 8.644881762761940981761216626260, 9.612263467989309009561866324283, 9.952229940815430873788554411728, 12.10514503037445088491348565762
