| L(s) = 1 | + (14.2 − 8.21i)2-s + (7 − 12.1i)4-s + (−369. − 213. i)5-s + (339.5 + 588. i)7-s + 3.97e3i·8-s − 7.02e3·10-s + (1.15e4 − 6.68e3i)11-s + (1.54e4 − 2.66e4i)13-s + (9.66e3 + 5.57e3i)14-s + (3.44e4 + 5.96e4i)16-s − 1.28e5i·17-s − 1.38e5·19-s + (−5.17e3 + 2.99e3i)20-s + (1.09e5 − 1.90e5i)22-s + (−2.63e5 − 1.51e5i)23-s + ⋯ |
| L(s) = 1 | + (0.889 − 0.513i)2-s + (0.0273 − 0.0473i)4-s + (−0.591 − 0.341i)5-s + (0.141 + 0.244i)7-s + 0.970i·8-s − 0.702·10-s + (0.791 − 0.456i)11-s + (0.539 − 0.934i)13-s + (0.251 + 0.145i)14-s + (0.525 + 0.910i)16-s − 1.53i·17-s − 1.06·19-s + (−0.0323 + 0.0186i)20-s + (0.469 − 0.812i)22-s + (−0.939 − 0.542i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.30824 - 1.86837i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.30824 - 1.86837i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (-14.2 + 8.21i)T + (128 - 221. i)T^{2} \) |
| 5 | \( 1 + (369. + 213. i)T + (1.95e5 + 3.38e5i)T^{2} \) |
| 7 | \( 1 + (-339.5 - 588. i)T + (-2.88e6 + 4.99e6i)T^{2} \) |
| 11 | \( 1 + (-1.15e4 + 6.68e3i)T + (1.07e8 - 1.85e8i)T^{2} \) |
| 13 | \( 1 + (-1.54e4 + 2.66e4i)T + (-4.07e8 - 7.06e8i)T^{2} \) |
| 17 | \( 1 + 1.28e5iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 1.38e5T + 1.69e10T^{2} \) |
| 23 | \( 1 + (2.63e5 + 1.51e5i)T + (3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 + (-1.14e6 + 6.63e5i)T + (2.50e11 - 4.33e11i)T^{2} \) |
| 31 | \( 1 + (1.76e5 - 3.05e5i)T + (-4.26e11 - 7.38e11i)T^{2} \) |
| 37 | \( 1 - 1.18e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + (-9.47e5 - 5.47e5i)T + (3.99e12 + 6.91e12i)T^{2} \) |
| 43 | \( 1 + (3.12e6 + 5.40e6i)T + (-5.84e12 + 1.01e13i)T^{2} \) |
| 47 | \( 1 + (2.07e3 - 1.19e3i)T + (1.19e13 - 2.06e13i)T^{2} \) |
| 53 | \( 1 - 1.25e7iT - 6.22e13T^{2} \) |
| 59 | \( 1 + (-9.12e6 - 5.26e6i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (8.29e6 + 1.43e7i)T + (-9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (3.83e6 - 6.63e6i)T + (-2.03e14 - 3.51e14i)T^{2} \) |
| 71 | \( 1 + 2.32e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 2.49e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + (2.08e7 + 3.61e7i)T + (-7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 + (-3.87e7 + 2.23e7i)T + (1.12e15 - 1.95e15i)T^{2} \) |
| 89 | \( 1 + 7.40e5iT - 3.93e15T^{2} \) |
| 97 | \( 1 + (-5.29e7 - 9.17e7i)T + (-3.91e15 + 6.78e15i)T^{2} \) |
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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
−12.17465314356873131416263975133, −11.77735239391146095788677343309, −10.51908586165719362501082992765, −8.797931688745386639764305901081, −7.994421957135508808422120148921, −6.17463424397504303131461065867, −4.79383757764169278146447961372, −3.79472504062666577731914285240, −2.49797371715962051857911649742, −0.53975931717687800411729175259,
1.47891338561664454816454237381, 3.75996057875769864480631292602, 4.40189924388686808187397232188, 6.10941404190216038816258023715, 6.87926519590918401010984935401, 8.325685955688129162684320189524, 9.766893378885456778840289645258, 11.00569131380935476649582864373, 12.17883617852109944588319658910, 13.19708493498120328350198820626
