| L(s) = 1 | + (76.6 + 26.2i)3-s + (−461. − 420. i)5-s + 982. i·7-s + (5.18e3 + 4.02e3i)9-s + 8.65e3i·11-s − 1.81e4i·13-s + (−2.43e4 − 4.43e4i)15-s − 4.88e3·17-s − 7.35e4·19-s + (−2.57e4 + 7.52e4i)21-s + 1.16e4·23-s + (3.61e4 + 3.88e5i)25-s + (2.91e5 + 4.44e5i)27-s + 9.81e5i·29-s − 8.53e5·31-s + ⋯ |
| L(s) = 1 | + (0.945 + 0.324i)3-s + (−0.739 − 0.673i)5-s + 0.409i·7-s + (0.789 + 0.613i)9-s + 0.591i·11-s − 0.636i·13-s + (−0.480 − 0.876i)15-s − 0.0584·17-s − 0.564·19-s + (−0.132 + 0.386i)21-s + 0.0416·23-s + (0.0925 + 0.995i)25-s + (0.548 + 0.836i)27-s + 1.38i·29-s − 0.923·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.480 - 0.876i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.480 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.538215892\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.538215892\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-76.6 - 26.2i)T \) |
| 5 | \( 1 + (461. + 420. i)T \) |
| good | 7 | \( 1 - 982. iT - 5.76e6T^{2} \) |
| 11 | \( 1 - 8.65e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 1.81e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + 4.88e3T + 6.97e9T^{2} \) |
| 19 | \( 1 + 7.35e4T + 1.69e10T^{2} \) |
| 23 | \( 1 - 1.16e4T + 7.83e10T^{2} \) |
| 29 | \( 1 - 9.81e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 8.53e5T + 8.52e11T^{2} \) |
| 37 | \( 1 - 7.12e5iT - 3.51e12T^{2} \) |
| 41 | \( 1 - 4.65e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 2.20e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + 3.60e6T + 2.38e13T^{2} \) |
| 53 | \( 1 + 1.28e7T + 6.22e13T^{2} \) |
| 59 | \( 1 - 1.13e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 8.48e6T + 1.91e14T^{2} \) |
| 67 | \( 1 - 9.68e6iT - 4.06e14T^{2} \) |
| 71 | \( 1 - 1.68e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 4.32e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 + 1.15e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + 7.18e7T + 2.25e15T^{2} \) |
| 89 | \( 1 - 7.55e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 1.21e8iT - 7.83e15T^{2} \) |
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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
−12.53891109440272185869511330419, −11.18806007591506491414463978212, −9.969120219108906222029443400659, −8.910310151679254691055258562751, −8.172997464393370728436417426108, −7.11682888228268350444035655137, −5.22915767032966273863955031821, −4.17515123363675537033360557964, −2.96371553798615270819705479058, −1.47914230915502662358226968975,
0.35366243094818097896241605708, 2.04744073338118047399287897667, 3.35480270910423979762424413566, 4.25724337746561790921179076835, 6.34860550429550680025465889135, 7.34244809109214147661687651007, 8.207429118070725036808055490519, 9.301907914914768967881096688832, 10.55934751456455935060522374455, 11.54960606482149675262128204278
