| L(s) = 1 | + (−1.50 − 0.852i)3-s + (−2.12 + 0.375i)5-s + (2.50 − 2.98i)7-s + (1.54 + 2.57i)9-s + (0.0357 − 0.202i)11-s + (−2.01 − 0.733i)13-s + (3.52 + 1.24i)15-s + (−5.87 + 3.38i)17-s + (−6.56 − 3.78i)19-s + (−6.31 + 2.36i)21-s + (−3.99 + 3.34i)23-s + (−0.312 + 0.113i)25-s + (−0.141 − 5.19i)27-s + (−0.709 − 1.94i)29-s + (−1.02 − 1.22i)31-s + ⋯ |
| L(s) = 1 | + (−0.870 − 0.492i)3-s + (−0.951 + 0.167i)5-s + (0.945 − 1.12i)7-s + (0.515 + 0.856i)9-s + (0.0107 − 0.0611i)11-s + (−0.558 − 0.203i)13-s + (0.910 + 0.322i)15-s + (−1.42 + 0.822i)17-s + (−1.50 − 0.868i)19-s + (−1.37 + 0.515i)21-s + (−0.832 + 0.698i)23-s + (−0.0624 + 0.0227i)25-s + (−0.0271 − 0.999i)27-s + (−0.131 − 0.361i)29-s + (−0.184 − 0.219i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0309i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0309i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00381650 + 0.246196i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00381650 + 0.246196i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.50 + 0.852i)T \) |
| good | 5 | \( 1 + (2.12 - 0.375i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-2.50 + 2.98i)T + (-1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (-0.0357 + 0.202i)T + (-10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (2.01 + 0.733i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (5.87 - 3.38i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (6.56 + 3.78i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.99 - 3.34i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (0.709 + 1.94i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (1.02 + 1.22i)T + (-5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (-3.90 - 6.75i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.12 + 11.3i)T + (-31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (6.15 + 1.08i)T + (40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-3.44 - 2.89i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 - 6.89iT - 53T^{2} \) |
| 59 | \( 1 + (-0.0937 - 0.531i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-0.695 - 0.583i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-1.77 + 4.88i)T + (-51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (5.04 + 8.74i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.46 + 2.53i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.25 + 3.45i)T + (-60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (10.0 - 3.66i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (9.79 + 5.65i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.67 + 9.50i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−11.00267193380425215005210073331, −10.21260890707188023229538796698, −8.581283907975423667950819888844, −7.70463967601838982479970321871, −7.11556831797442111802419613820, −6.07770734909977413230446652313, −4.59949626849635235899526775293, −4.12899641736425384536143559895, −1.96590772931186545438419439459, −0.16632714905817582154733836339,
2.21405844213623478761084205666, 4.15031686029002255159453623502, 4.71805231534941936339843079076, 5.82196195372827518137180410187, 6.85529252761352481620651172758, 8.117313213902878794861351525265, 8.807817483198856474625852784935, 9.882431047102598632213133816459, 11.03767760808317517872584188265, 11.52347545496574238304743852412
