| L(s) = 1 | + (1.40 + 0.136i)2-s + (−1.71 + 0.213i)3-s + (1.96 + 0.384i)4-s + (0.362 − 1.82i)5-s + (−2.44 + 0.0661i)6-s + (1.97 + 0.817i)7-s + (2.71 + 0.809i)8-s + (2.90 − 0.734i)9-s + (0.760 − 2.51i)10-s + (1.09 + 1.64i)11-s + (−3.45 − 0.241i)12-s + (−0.507 + 0.100i)13-s + (2.66 + 1.42i)14-s + (−0.233 + 3.21i)15-s + (3.70 + 1.50i)16-s + (−3.76 − 3.76i)17-s + ⋯ |
| L(s) = 1 | + (0.995 + 0.0965i)2-s + (−0.992 + 0.123i)3-s + (0.981 + 0.192i)4-s + (0.162 − 0.815i)5-s + (−0.999 + 0.0270i)6-s + (0.746 + 0.309i)7-s + (0.958 + 0.286i)8-s + (0.969 − 0.244i)9-s + (0.240 − 0.796i)10-s + (0.330 + 0.494i)11-s + (−0.997 − 0.0696i)12-s + (−0.140 + 0.0279i)13-s + (0.712 + 0.379i)14-s + (−0.0603 + 0.829i)15-s + (0.926 + 0.377i)16-s + (−0.914 − 0.914i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0341i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0341i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.72054 + 0.0294022i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.72054 + 0.0294022i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.40 - 0.136i)T \) |
| 3 | \( 1 + (1.71 - 0.213i)T \) |
| good | 5 | \( 1 + (-0.362 + 1.82i)T + (-4.61 - 1.91i)T^{2} \) |
| 7 | \( 1 + (-1.97 - 0.817i)T + (4.94 + 4.94i)T^{2} \) |
| 11 | \( 1 + (-1.09 - 1.64i)T + (-4.20 + 10.1i)T^{2} \) |
| 13 | \( 1 + (0.507 - 0.100i)T + (12.0 - 4.97i)T^{2} \) |
| 17 | \( 1 + (3.76 + 3.76i)T + 17iT^{2} \) |
| 19 | \( 1 + (7.63 - 1.51i)T + (17.5 - 7.27i)T^{2} \) |
| 23 | \( 1 + (1.55 - 0.645i)T + (16.2 - 16.2i)T^{2} \) |
| 29 | \( 1 + (4.87 + 3.25i)T + (11.0 + 26.7i)T^{2} \) |
| 31 | \( 1 - 3.85T + 31T^{2} \) |
| 37 | \( 1 + (-0.101 + 0.508i)T + (-34.1 - 14.1i)T^{2} \) |
| 41 | \( 1 + (-1.45 - 3.51i)T + (-28.9 + 28.9i)T^{2} \) |
| 43 | \( 1 + (3.41 + 5.11i)T + (-16.4 + 39.7i)T^{2} \) |
| 47 | \( 1 + (7.48 - 7.48i)T - 47iT^{2} \) |
| 53 | \( 1 + (7.62 - 5.09i)T + (20.2 - 48.9i)T^{2} \) |
| 59 | \( 1 + (-3.71 - 0.739i)T + (54.5 + 22.5i)T^{2} \) |
| 61 | \( 1 + (-11.6 - 7.75i)T + (23.3 + 56.3i)T^{2} \) |
| 67 | \( 1 + (-7.63 + 11.4i)T + (-25.6 - 61.8i)T^{2} \) |
| 71 | \( 1 + (0.709 - 1.71i)T + (-50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (-4.01 - 9.68i)T + (-51.6 + 51.6i)T^{2} \) |
| 79 | \( 1 + (7.75 - 7.75i)T - 79iT^{2} \) |
| 83 | \( 1 + (3.08 + 15.5i)T + (-76.6 + 31.7i)T^{2} \) |
| 89 | \( 1 + (-2.28 + 5.51i)T + (-62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + 12.5iT - 97T^{2} \) |
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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
−12.60411373504331721423376932960, −11.66165862722153219444603744076, −11.05779182314728803595849297240, −9.784475192337631009967721361263, −8.382143335437338810493652202008, −7.01815527431729933240591974825, −6.00102000604944265181991069476, −4.88555049365419814941754627239, −4.32257223779425077022772096039, −1.90404014992104729005739875797,
2.00377865469765471097829117965, 3.93718832027639009710498085451, 4.98735203162195535935653714131, 6.34699679270480816983955858237, 6.76548726682404685852759384843, 8.208953993938344909285840599853, 10.20175249739807719110312045812, 11.00215670449441439841923902247, 11.33731404266149555818837614639, 12.62452742521427400417330354006
