| L(s) = 1 | − 11.3·2-s + 46.7i·3-s + 128.·4-s − 1.21e3i·5-s − 529. i·6-s + (1.53e3 + 1.85e3i)7-s − 1.44e3·8-s − 2.18e3·9-s + 1.37e4i·10-s − 6.83e3·11-s + 5.98e3i·12-s + 1.77e4i·13-s + (−1.73e4 − 2.09e4i)14-s + 5.69e4·15-s + 1.63e4·16-s − 7.47e4i·17-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577i·3-s + 0.500·4-s − 1.94i·5-s − 0.408i·6-s + (0.637 + 0.770i)7-s − 0.353·8-s − 0.333·9-s + 1.37i·10-s − 0.466·11-s + 0.288i·12-s + 0.623i·13-s + (−0.450 − 0.544i)14-s + 1.12·15-s + 0.250·16-s − 0.895i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.770 + 0.637i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.770 + 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.198547 - 0.551490i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.198547 - 0.551490i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 11.3T \) |
| 3 | \( 1 - 46.7iT \) |
| 7 | \( 1 + (-1.53e3 - 1.85e3i)T \) |
| good | 5 | \( 1 + 1.21e3iT - 3.90e5T^{2} \) |
| 11 | \( 1 + 6.83e3T + 2.14e8T^{2} \) |
| 13 | \( 1 - 1.77e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + 7.47e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 7.50e4iT - 1.69e10T^{2} \) |
| 23 | \( 1 + 4.11e5T + 7.83e10T^{2} \) |
| 29 | \( 1 - 2.38e5T + 5.00e11T^{2} \) |
| 31 | \( 1 + 1.56e6iT - 8.52e11T^{2} \) |
| 37 | \( 1 + 3.44e6T + 3.51e12T^{2} \) |
| 41 | \( 1 - 8.01e5iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 2.39e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + 3.96e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 8.82e6T + 6.22e13T^{2} \) |
| 59 | \( 1 + 1.19e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 6.09e6iT - 1.91e14T^{2} \) |
| 67 | \( 1 + 2.76e7T + 4.06e14T^{2} \) |
| 71 | \( 1 + 8.48e6T + 6.45e14T^{2} \) |
| 73 | \( 1 - 1.38e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 - 4.60e7T + 1.51e15T^{2} \) |
| 83 | \( 1 - 7.71e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 3.73e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 1.35e7iT - 7.83e15T^{2} \) |
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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
−13.78785992460355653654870624257, −12.30220323367851687137946761581, −11.49485118311993234409457487824, −9.701784413305692960605659377404, −8.888700005565506492997790374379, −7.997883413569091959042389583503, −5.59523430015502219688295618197, −4.52401488563609333711691818685, −1.95005557045804462401331159080, −0.26841445535521899606169678880,
1.83360804161583418563423225094, 3.34917538726892715585274155116, 6.13495406484733715243786425246, 7.28736142291895745859446482650, 8.087712677111518441025717131488, 10.41751111183242068980272590096, 10.62954655922615236893033723872, 12.08152229877127436800170477462, 13.87877108514525562969024921047, 14.60626407769845408480524530497
