| L(s) = 1 | − 2.20·3-s + (3.56 + 3.50i)5-s + 0.620·7-s − 4.12·9-s − 17.6i·11-s + 17.9i·13-s + (−7.86 − 7.74i)15-s − 10.9i·17-s + 2.17i·19-s − 1.36·21-s + 13.8·23-s + (0.369 + 24.9i)25-s + 28.9·27-s − 14.7·29-s + 55.2i·31-s + ⋯ |
| L(s) = 1 | − 0.736·3-s + (0.712 + 0.701i)5-s + 0.0885·7-s − 0.458·9-s − 1.60i·11-s + 1.38i·13-s + (−0.524 − 0.516i)15-s − 0.644i·17-s + 0.114i·19-s − 0.0652·21-s + 0.603·23-s + (0.0147 + 0.999i)25-s + 1.07·27-s − 0.508·29-s + 1.78i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.712 - 0.701i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.712 - 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7941562991\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7941562991\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-3.56 - 3.50i)T \) |
| good | 3 | \( 1 + 2.20T + 9T^{2} \) |
| 7 | \( 1 - 0.620T + 49T^{2} \) |
| 11 | \( 1 + 17.6iT - 121T^{2} \) |
| 13 | \( 1 - 17.9iT - 169T^{2} \) |
| 17 | \( 1 + 10.9iT - 289T^{2} \) |
| 19 | \( 1 - 2.17iT - 361T^{2} \) |
| 23 | \( 1 - 13.8T + 529T^{2} \) |
| 29 | \( 1 + 14.7T + 841T^{2} \) |
| 31 | \( 1 - 55.2iT - 961T^{2} \) |
| 37 | \( 1 - 7.01iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 48.3T + 1.68e3T^{2} \) |
| 43 | \( 1 - 58.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + 54.1T + 2.20e3T^{2} \) |
| 53 | \( 1 - 3.94iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 72.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 79.3T + 3.72e3T^{2} \) |
| 67 | \( 1 + 26.7T + 4.48e3T^{2} \) |
| 71 | \( 1 - 46.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 95.1iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 24.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 107.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 44.2T + 7.92e3T^{2} \) |
| 97 | \( 1 - 123. iT - 9.40e3T^{2} \) |
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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
−9.823610690792541230750017353688, −8.993115537584409962223450224734, −8.324534025770226700039962402915, −6.90848585191496883398861248347, −6.54202373237024051432689003592, −5.62337530303483418511449881847, −5.00903852568872085978414356204, −3.54540560065071287747453050138, −2.68805500460501254560246020247, −1.29317850297521565108300598319,
0.26129752981871708298758334664, 1.59904239855433504865643905407, 2.72685879155137577876025963381, 4.23753980484772126684300284630, 5.12570449948028391485573168419, 5.67113874880309962380480477799, 6.49763092679346792561591540683, 7.57799479401095569761826704249, 8.353256426985184391009829187708, 9.322869639619373083594708489888
