L(s) = 1 | + (−1.96 + 0.384i)2-s + (−1.10 − 2.67i)3-s + (3.70 − 1.50i)4-s + (2.95 − 7.13i)5-s + (3.20 + 4.82i)6-s + (−4.18 + 4.18i)7-s + (−6.69 + 4.38i)8-s + (0.437 − 0.437i)9-s + (−3.05 + 15.1i)10-s + (1.42 − 3.44i)11-s + (−8.13 − 8.23i)12-s + (8.39 + 20.2i)13-s + (6.60 − 9.82i)14-s − 22.3·15-s + (11.4 − 11.1i)16-s − 1.73i·17-s + ⋯ |
L(s) = 1 | + (−0.981 + 0.192i)2-s + (−0.369 − 0.891i)3-s + (0.926 − 0.377i)4-s + (0.591 − 1.42i)5-s + (0.533 + 0.803i)6-s + (−0.597 + 0.597i)7-s + (−0.836 + 0.547i)8-s + (0.0486 − 0.0486i)9-s + (−0.305 + 1.51i)10-s + (0.129 − 0.313i)11-s + (−0.678 − 0.686i)12-s + (0.646 + 1.55i)13-s + (0.471 − 0.701i)14-s − 1.49·15-s + (0.715 − 0.698i)16-s − 0.101i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.368 + 0.929i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.368 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.538414 - 0.365704i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.538414 - 0.365704i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.96 - 0.384i)T \) |
good | 3 | \( 1 + (1.10 + 2.67i)T + (-6.36 + 6.36i)T^{2} \) |
| 5 | \( 1 + (-2.95 + 7.13i)T + (-17.6 - 17.6i)T^{2} \) |
| 7 | \( 1 + (4.18 - 4.18i)T - 49iT^{2} \) |
| 11 | \( 1 + (-1.42 + 3.44i)T + (-85.5 - 85.5i)T^{2} \) |
| 13 | \( 1 + (-8.39 - 20.2i)T + (-119. + 119. i)T^{2} \) |
| 17 | \( 1 + 1.73iT - 289T^{2} \) |
| 19 | \( 1 + (-14.2 + 5.90i)T + (255. - 255. i)T^{2} \) |
| 23 | \( 1 + (-15.1 - 15.1i)T + 529iT^{2} \) |
| 29 | \( 1 + (6.74 - 2.79i)T + (594. - 594. i)T^{2} \) |
| 31 | \( 1 - 31.1iT - 961T^{2} \) |
| 37 | \( 1 + (-5.30 + 12.7i)T + (-968. - 968. i)T^{2} \) |
| 41 | \( 1 + (18.5 - 18.5i)T - 1.68e3iT^{2} \) |
| 43 | \( 1 + (-31.0 + 75.0i)T + (-1.30e3 - 1.30e3i)T^{2} \) |
| 47 | \( 1 + 16.2T + 2.20e3T^{2} \) |
| 53 | \( 1 + (29.0 + 12.0i)T + (1.98e3 + 1.98e3i)T^{2} \) |
| 59 | \( 1 + (-34.1 - 14.1i)T + (2.46e3 + 2.46e3i)T^{2} \) |
| 61 | \( 1 + (68.7 - 28.4i)T + (2.63e3 - 2.63e3i)T^{2} \) |
| 67 | \( 1 + (-10.5 - 25.3i)T + (-3.17e3 + 3.17e3i)T^{2} \) |
| 71 | \( 1 + (32.2 - 32.2i)T - 5.04e3iT^{2} \) |
| 73 | \( 1 + (28.5 - 28.5i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 - 22.4T + 6.24e3T^{2} \) |
| 83 | \( 1 + (123. - 51.0i)T + (4.87e3 - 4.87e3i)T^{2} \) |
| 89 | \( 1 + (-61.0 - 61.0i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + 69.9T + 9.40e3T^{2} \) |
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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
−16.58531626773983365378056467132, −15.78271057940164696650893934198, −13.71743788146355851047955659436, −12.50348740816206239312292494807, −11.56568717291843162517662250005, −9.454840873286585218640004588803, −8.806568014852890017662309142447, −6.93753023002268434798105583392, −5.70198562050249527448764124426, −1.40187753485139279992189654609,
3.23418827618785263716348558489, 6.14867038322074340446203340805, 7.55661083544069518792700062743, 9.719250319171774281156642125680, 10.37991085298396773578965333873, 11.10732771863122502140357155622, 13.11483173561296725178847355702, 14.87092447575565937491463033506, 15.81370320247183074366420526705, 16.94785903845440224944527015812