| L(s) = 1 | + (4.33 + 2.5i)2-s + (−2.59 + 1.5i)3-s + (8.50 + 14.7i)4-s + (8.52 + 7.23i)5-s − 15.0·6-s + (−15.5 − 10i)7-s + 45.0i·8-s + (4.5 − 7.79i)9-s + (18.8 + 52.6i)10-s + (22.5 + 38.9i)11-s + (−44.1 − 25.5i)12-s − 37i·13-s + (−42.5 − 82.2i)14-s + (−33 − 6i)15-s + (−44.5 + 77.0i)16-s + (−50.2 + 29i)17-s + ⋯ |
| L(s) = 1 | + (1.53 + 0.883i)2-s + (−0.499 + 0.288i)3-s + (1.06 + 1.84i)4-s + (0.762 + 0.646i)5-s − 1.02·6-s + (−0.841 − 0.539i)7-s + 1.98i·8-s + (0.166 − 0.288i)9-s + (0.595 + 1.66i)10-s + (0.616 + 1.06i)11-s + (−1.06 − 0.613i)12-s − 0.789i·13-s + (−0.811 − 1.57i)14-s + (−0.568 − 0.103i)15-s + (−0.695 + 1.20i)16-s + (−0.716 + 0.413i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.352 - 0.935i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.352 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.81923 + 2.62956i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.81923 + 2.62956i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.59 - 1.5i)T \) |
| 5 | \( 1 + (-8.52 - 7.23i)T \) |
| 7 | \( 1 + (15.5 + 10i)T \) |
| good | 2 | \( 1 + (-4.33 - 2.5i)T + (4 + 6.92i)T^{2} \) |
| 11 | \( 1 + (-22.5 - 38.9i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 37iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (50.2 - 29i)T + (2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-67.5 + 116. i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-54.5 - 31.5i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 - 184T + 2.43e4T^{2} \) |
| 31 | \( 1 + (85 + 147. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (61.4 + 35.5i)T + (2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 - 247T + 6.89e4T^{2} \) |
| 43 | \( 1 + 242iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (491. + 283.5i)T + (5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-38.9 + 22.5i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (2 + 3.46i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (131 - 226. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (756. - 437i)T + (1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 1.13e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + (93.5 - 54i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (31 - 53.6i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 454iT - 5.71e5T^{2} \) |
| 89 | \( 1 + (603 - 1.04e3i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 1.36e3iT - 9.12e5T^{2} \) |
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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
−13.48565186179624389594592261361, −13.04666834290447744040696242561, −11.84966006371205878734076186880, −10.58660130244003178817900574131, −9.436421170167226449409314878336, −7.19912836814955642284616698659, −6.66974195148953556916776951463, −5.57495047385299272744789853284, −4.35099725222925725326566953064, −2.97703928999345220578159389220,
1.43721061311397685907556067212, 3.07214498318776039233368851756, 4.65843082667287493475857326712, 5.86183798865195934209940777078, 6.45075123315215320871426222173, 8.936878366601372895663272676518, 10.08220381630524291452647244349, 11.33557353350552465949286986995, 12.14980731509198085356267317325, 12.85792666987429914733366310091
