| L(s) = 1 | + (2.88 − 0.812i)3-s + (0.980 − 1.16i)5-s + (3.23 + 1.17i)7-s + (7.67 − 4.69i)9-s + (−3.21 − 3.82i)11-s + (−0.778 + 4.41i)13-s + (1.88 − 4.17i)15-s + (−3.57 − 2.06i)17-s + (6.75 + 11.7i)19-s + (10.3 + 0.772i)21-s + (−5.79 − 15.9i)23-s + (3.93 + 22.3i)25-s + (18.3 − 19.7i)27-s + (−47.1 + 8.30i)29-s + (−14.3 + 5.22i)31-s + ⋯ |
| L(s) = 1 | + (0.962 − 0.270i)3-s + (0.196 − 0.233i)5-s + (0.462 + 0.168i)7-s + (0.853 − 0.521i)9-s + (−0.291 − 0.347i)11-s + (−0.0599 + 0.339i)13-s + (0.125 − 0.278i)15-s + (−0.210 − 0.121i)17-s + (0.355 + 0.615i)19-s + (0.491 + 0.0367i)21-s + (−0.252 − 0.692i)23-s + (0.157 + 0.893i)25-s + (0.680 − 0.733i)27-s + (−1.62 + 0.286i)29-s + (−0.462 + 0.168i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.334i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.942 + 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.84449 - 0.317250i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.84449 - 0.317250i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.88 + 0.812i)T \) |
| good | 5 | \( 1 + (-0.980 + 1.16i)T + (-4.34 - 24.6i)T^{2} \) |
| 7 | \( 1 + (-3.23 - 1.17i)T + (37.5 + 31.4i)T^{2} \) |
| 11 | \( 1 + (3.21 + 3.82i)T + (-21.0 + 119. i)T^{2} \) |
| 13 | \( 1 + (0.778 - 4.41i)T + (-158. - 57.8i)T^{2} \) |
| 17 | \( 1 + (3.57 + 2.06i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-6.75 - 11.7i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (5.79 + 15.9i)T + (-405. + 340. i)T^{2} \) |
| 29 | \( 1 + (47.1 - 8.30i)T + (790. - 287. i)T^{2} \) |
| 31 | \( 1 + (14.3 - 5.22i)T + (736. - 617. i)T^{2} \) |
| 37 | \( 1 + (32.3 - 56.0i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (55.4 + 9.76i)T + (1.57e3 + 574. i)T^{2} \) |
| 43 | \( 1 + (-22.7 + 19.0i)T + (321. - 1.82e3i)T^{2} \) |
| 47 | \( 1 + (7.04 - 19.3i)T + (-1.69e3 - 1.41e3i)T^{2} \) |
| 53 | \( 1 - 19.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-63.6 + 75.9i)T + (-604. - 3.42e3i)T^{2} \) |
| 61 | \( 1 + (-77.5 - 28.2i)T + (2.85e3 + 2.39e3i)T^{2} \) |
| 67 | \( 1 + (-11.2 + 63.7i)T + (-4.21e3 - 1.53e3i)T^{2} \) |
| 71 | \( 1 + (-109. - 63.3i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-18.0 - 31.2i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (14.4 + 82.1i)T + (-5.86e3 + 2.13e3i)T^{2} \) |
| 83 | \( 1 + (16.5 - 2.91i)T + (6.47e3 - 2.35e3i)T^{2} \) |
| 89 | \( 1 + (-66.0 + 38.1i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-82.1 + 68.9i)T + (1.63e3 - 9.26e3i)T^{2} \) |
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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
−13.48196893112784938829081487272, −12.58507263798865248296985372965, −11.38549854991031435064195609625, −10.01265812654839720307539157072, −8.940453468885260607868061260806, −8.066699056910992685855513005810, −6.87042163097813747884371921444, −5.21326135508421548416794106201, −3.55219831524948682524962395281, −1.84102769588108979564286071308,
2.17777231796777701186422145520, 3.78685313200206169813840859474, 5.24755113741049135973818483433, 7.09909253637948803491141467294, 8.054359842723758777633652211078, 9.236474177103108031915304470255, 10.20663007001672647677499985586, 11.25808143416264390947548778710, 12.75088610665459052692092916520, 13.63361960857972212927145520074
