| L(s) = 1 | + (1.78 + 3.09i)2-s + (1.34 + 15.5i)3-s + (9.59 − 16.6i)4-s + (4.05 − 7.02i)5-s + (−45.7 + 31.9i)6-s + (−87.7 − 151. i)7-s + 183.·8-s + (−239. + 41.6i)9-s + 29.0·10-s + (206. + 358. i)11-s + (270. + 126. i)12-s + (−61.8 + 107. i)13-s + (313. − 543. i)14-s + (114. + 53.5i)15-s + (20.9 + 36.2i)16-s − 2.22e3·17-s + ⋯ |
| L(s) = 1 | + (0.316 + 0.548i)2-s + (0.0860 + 0.996i)3-s + (0.299 − 0.519i)4-s + (0.0725 − 0.125i)5-s + (−0.518 + 0.362i)6-s + (−0.676 − 1.17i)7-s + 1.01·8-s + (−0.985 + 0.171i)9-s + 0.0917·10-s + (0.515 + 0.892i)11-s + (0.543 + 0.253i)12-s + (−0.101 + 0.175i)13-s + (0.428 − 0.741i)14-s + (0.131 + 0.0614i)15-s + (0.0204 + 0.0354i)16-s − 1.86·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.644 - 0.764i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.644 - 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.23811 + 0.575718i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.23811 + 0.575718i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.34 - 15.5i)T \) |
| good | 2 | \( 1 + (-1.78 - 3.09i)T + (-16 + 27.7i)T^{2} \) |
| 5 | \( 1 + (-4.05 + 7.02i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 + (87.7 + 151. i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + (-206. - 358. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + (61.8 - 107. i)T + (-1.85e5 - 3.21e5i)T^{2} \) |
| 17 | \( 1 + 2.22e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 891.T + 2.47e6T^{2} \) |
| 23 | \( 1 + (-293. + 509. i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + (-2.46e3 - 4.27e3i)T + (-1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 + (-2.83e3 + 4.90e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 - 1.20e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (1.19e3 - 2.06e3i)T + (-5.79e7 - 1.00e8i)T^{2} \) |
| 43 | \( 1 + (-2.88e3 - 4.99e3i)T + (-7.35e7 + 1.27e8i)T^{2} \) |
| 47 | \( 1 + (3.53e3 + 6.11e3i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + 1.15e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + (4.87e3 - 8.44e3i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-932. - 1.61e3i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-1.96e4 + 3.40e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 - 4.36e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.21e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (2.53e4 + 4.39e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-2.78e4 - 4.81e4i)T + (-1.96e9 + 3.41e9i)T^{2} \) |
| 89 | \( 1 + 4.65e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (-2.16e4 - 3.74e4i)T + (-4.29e9 + 7.43e9i)T^{2} \) |
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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
−20.34586857494468206156782259547, −19.76095872550773885023726569210, −17.19126519725055994301013277831, −16.08630971026167262790584150834, −14.91713017427128862489185646305, −13.56544380571294073754861973020, −10.92502240575137764061910331642, −9.599936054004866068769170014839, −6.80264052886584769382324922843, −4.52107944278542293103835939860,
2.69186648862284435527492701799, 6.50441568911839748830159418715, 8.620783377185363944361770524520, 11.40920211873309176949080305217, 12.51519167909506161386901175195, 13.72766459426378531151663894498, 15.86857811536595348047267456988, 17.59659137975063919261899024717, 19.05471966234483361515029387629, 20.03791263409129928036533220474
