| L(s) = 1 | − 1.73·3-s + (−4.27 − 2.59i)5-s + 0.837·7-s + 2.99·9-s − 15.7i·11-s + 5.18i·13-s + (7.40 + 4.49i)15-s − 27.3i·17-s + 17.9i·19-s − 1.45·21-s − 19.1·23-s + (11.5 + 22.1i)25-s − 5.19·27-s + 45.6·29-s − 13.6i·31-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + (−0.854 − 0.518i)5-s + 0.119·7-s + 0.333·9-s − 1.43i·11-s + 0.398i·13-s + (0.493 + 0.299i)15-s − 1.60i·17-s + 0.945i·19-s − 0.0690·21-s − 0.830·23-s + (0.461 + 0.886i)25-s − 0.192·27-s + 1.57·29-s − 0.439i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.854 - 0.518i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.854 - 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1239584946\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1239584946\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73T \) |
| 5 | \( 1 + (4.27 + 2.59i)T \) |
| good | 7 | \( 1 - 0.837T + 49T^{2} \) |
| 11 | \( 1 + 15.7iT - 121T^{2} \) |
| 13 | \( 1 - 5.18iT - 169T^{2} \) |
| 17 | \( 1 + 27.3iT - 289T^{2} \) |
| 19 | \( 1 - 17.9iT - 361T^{2} \) |
| 23 | \( 1 + 19.1T + 529T^{2} \) |
| 29 | \( 1 - 45.6T + 841T^{2} \) |
| 31 | \( 1 + 13.6iT - 961T^{2} \) |
| 37 | \( 1 + 15.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 13.2T + 1.68e3T^{2} \) |
| 43 | \( 1 + 27.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + 55.6T + 2.20e3T^{2} \) |
| 53 | \( 1 + 15.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 87.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 38T + 3.72e3T^{2} \) |
| 67 | \( 1 + 92.2T + 4.48e3T^{2} \) |
| 71 | \( 1 - 130. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 54.7iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 13.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 59.0T + 6.88e3T^{2} \) |
| 89 | \( 1 - 39.8T + 7.92e3T^{2} \) |
| 97 | \( 1 - 168. iT - 9.40e3T^{2} \) |
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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
−9.297147612017778744272863416392, −8.401132197499851691516925229971, −7.78251605864011389351525522131, −6.74066258306610225598698286739, −5.81804039668772689813040353168, −4.92571065849843056458041354143, −4.04061733237711833512104368735, −2.96625263476086775497748392040, −1.15757574447309014126204355160, −0.04857835803796854480342185229,
1.68654242179898356822947790841, 3.09347233109709529023541219935, 4.28712369632386514003786992840, 4.87684012676824255535799025813, 6.26674926908928759083998764892, 6.83724922778901603144018962154, 7.80520812661276864059058482321, 8.424228702261911872810706333997, 9.754561623288557886492261947605, 10.40643835069912489952116468135
