| L(s) = 1 | + (5.00 + 5.96i)5-s + (−3.39 + 1.23i)7-s + (−2.59 + 3.09i)11-s + (2.31 + 13.1i)13-s + (−20.7 + 11.9i)17-s + (13.5 − 23.5i)19-s + (−3.97 + 10.9i)23-s + (−6.17 + 35.0i)25-s + (−22.8 − 4.03i)29-s + (−3.81 − 1.38i)31-s + (−24.3 − 14.0i)35-s + (35.3 + 61.2i)37-s + (43.0 − 7.59i)41-s + (35.3 + 29.6i)43-s + (28.3 + 77.9i)47-s + ⋯ |
| L(s) = 1 | + (1.00 + 1.19i)5-s + (−0.484 + 0.176i)7-s + (−0.236 + 0.281i)11-s + (0.177 + 1.00i)13-s + (−1.22 + 0.705i)17-s + (0.715 − 1.23i)19-s + (−0.172 + 0.474i)23-s + (−0.247 + 1.40i)25-s + (−0.788 − 0.139i)29-s + (−0.122 − 0.0447i)31-s + (−0.695 − 0.401i)35-s + (0.956 + 1.65i)37-s + (1.05 − 0.185i)41-s + (0.822 + 0.690i)43-s + (0.603 + 1.65i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.117 - 0.993i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.117 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.06024 + 1.19346i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.06024 + 1.19346i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-5.00 - 5.96i)T + (-4.34 + 24.6i)T^{2} \) |
| 7 | \( 1 + (3.39 - 1.23i)T + (37.5 - 31.4i)T^{2} \) |
| 11 | \( 1 + (2.59 - 3.09i)T + (-21.0 - 119. i)T^{2} \) |
| 13 | \( 1 + (-2.31 - 13.1i)T + (-158. + 57.8i)T^{2} \) |
| 17 | \( 1 + (20.7 - 11.9i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-13.5 + 23.5i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (3.97 - 10.9i)T + (-405. - 340. i)T^{2} \) |
| 29 | \( 1 + (22.8 + 4.03i)T + (790. + 287. i)T^{2} \) |
| 31 | \( 1 + (3.81 + 1.38i)T + (736. + 617. i)T^{2} \) |
| 37 | \( 1 + (-35.3 - 61.2i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-43.0 + 7.59i)T + (1.57e3 - 574. i)T^{2} \) |
| 43 | \( 1 + (-35.3 - 29.6i)T + (321. + 1.82e3i)T^{2} \) |
| 47 | \( 1 + (-28.3 - 77.9i)T + (-1.69e3 + 1.41e3i)T^{2} \) |
| 53 | \( 1 + 28.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (33.8 + 40.3i)T + (-604. + 3.42e3i)T^{2} \) |
| 61 | \( 1 + (-4.08 + 1.48i)T + (2.85e3 - 2.39e3i)T^{2} \) |
| 67 | \( 1 + (22.7 + 129. i)T + (-4.21e3 + 1.53e3i)T^{2} \) |
| 71 | \( 1 + (-60.4 + 34.8i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-65.4 + 113. i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (4.20 - 23.8i)T + (-5.86e3 - 2.13e3i)T^{2} \) |
| 83 | \( 1 + (-41.0 - 7.22i)T + (6.47e3 + 2.35e3i)T^{2} \) |
| 89 | \( 1 + (84.1 + 48.6i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-37.2 - 31.2i)T + (1.63e3 + 9.26e3i)T^{2} \) |
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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
−11.28146705139087066116893105334, −10.88204162514235759257895224841, −9.571644019749393337149150440834, −9.286747393505981563401184658113, −7.64235602579587232561241045083, −6.57533350881142188633359291083, −6.10895378620487641172195472122, −4.58193294741408262489882283867, −3.04654087844567705678270817340, −2.01245775982002575455105332524,
0.74474173489548966892160426895, 2.36438660868196149045577156217, 3.99614143024052861925414053064, 5.37758576315955162862565210597, 5.90957462871872048037226457560, 7.34333775753039735667351378669, 8.504267603753714954353311744094, 9.297649275731318016956184794143, 10.06107741015074289296299615587, 11.04328612510141421186978311404
