| L(s) = 1 | + (80.8 − 5.08i)3-s + (615. − 105. i)5-s − 2.67e3i·7-s + (6.50e3 − 822. i)9-s + 2.70e4i·11-s + 4.45e4i·13-s + (4.92e4 − 1.16e4i)15-s − 3.16e4·17-s + 2.04e5·19-s + (−1.36e4 − 2.16e5i)21-s − 1.80e4·23-s + (3.68e5 − 1.30e5i)25-s + (5.22e5 − 9.95e4i)27-s + 9.82e5i·29-s − 3.91e5·31-s + ⋯ |
| L(s) = 1 | + (0.998 − 0.0627i)3-s + (0.985 − 0.169i)5-s − 1.11i·7-s + (0.992 − 0.125i)9-s + 1.84i·11-s + 1.55i·13-s + (0.972 − 0.231i)15-s − 0.378·17-s + 1.57·19-s + (−0.0699 − 1.11i)21-s − 0.0643·23-s + (0.942 − 0.334i)25-s + (0.982 − 0.187i)27-s + 1.38i·29-s − 0.424·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.231i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.972 - 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(3.930476655\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.930476655\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-80.8 + 5.08i)T \) |
| 5 | \( 1 + (-615. + 105. i)T \) |
| good | 7 | \( 1 + 2.67e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 - 2.70e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 4.45e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + 3.16e4T + 6.97e9T^{2} \) |
| 19 | \( 1 - 2.04e5T + 1.69e10T^{2} \) |
| 23 | \( 1 + 1.80e4T + 7.83e10T^{2} \) |
| 29 | \( 1 - 9.82e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 3.91e5T + 8.52e11T^{2} \) |
| 37 | \( 1 + 2.72e6iT - 3.51e12T^{2} \) |
| 41 | \( 1 + 1.49e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 8.26e5iT - 1.16e13T^{2} \) |
| 47 | \( 1 + 3.41e3T + 2.38e13T^{2} \) |
| 53 | \( 1 - 1.00e7T + 6.22e13T^{2} \) |
| 59 | \( 1 + 1.36e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 1.72e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + 2.13e7iT - 4.06e14T^{2} \) |
| 71 | \( 1 - 3.02e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 2.74e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 - 1.01e7T + 1.51e15T^{2} \) |
| 83 | \( 1 - 4.89e7T + 2.25e15T^{2} \) |
| 89 | \( 1 + 6.65e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 7.67e7iT - 7.83e15T^{2} \) |
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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
−12.21874623225550378346198057524, −10.57126077931698074076536158324, −9.613487322725234657140057208033, −9.099847731618498979122961552022, −7.33077185664440184140680494204, −6.92599434283013378034996380858, −4.90804930551240492195413103017, −3.87956500417947720641807566275, −2.17912896008376266949613520243, −1.37077477089683912966234335761,
1.03441587406844214018904793242, 2.61543380805364569101173759604, 3.22305999972595590731834521989, 5.33322175082759160040587423789, 6.12574174638762232741596224565, 7.85482424473491338204532663505, 8.720892595943712090687043969281, 9.590456457795161856893372472140, 10.62008408880719919857217654813, 11.94089223867297442406531444454
