| L(s) = 1 | − 2-s − 2.82i·3-s + 4-s + 2.82i·5-s + 2.82i·6-s − 8-s − 5.00·9-s − 2.82i·10-s + 2.82i·11-s − 2.82i·12-s + 2·13-s + 8.00·15-s + 16-s + (−3 − 2.82i)17-s + 5.00·18-s − 4·19-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.63i·3-s + 0.5·4-s + 1.26i·5-s + 1.15i·6-s − 0.353·8-s − 1.66·9-s − 0.894i·10-s + 0.852i·11-s − 0.816i·12-s + 0.554·13-s + 2.06·15-s + 0.250·16-s + (−0.727 − 0.685i)17-s + 1.17·18-s − 0.917·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 34 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 + 0.685i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.727 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.516960 - 0.205273i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.516960 - 0.205273i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 17 | \( 1 + (3 + 2.82i)T \) |
| good | 3 | \( 1 + 2.82iT - 3T^{2} \) |
| 5 | \( 1 - 2.82iT - 5T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 2.82iT - 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 5.65iT - 23T^{2} \) |
| 29 | \( 1 - 2.82iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 8.48iT - 37T^{2} \) |
| 41 | \( 1 + 5.65iT - 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 8.48iT - 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 5.65iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 16.9iT - 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 16.9iT - 97T^{2} \) |
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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
−17.06167896434940375430658952132, −15.27285023183387587548225226051, −14.12719507516391158951572154901, −12.84960809136784896974761597774, −11.62104328433788920008879166606, −10.44767684414425161642458569644, −8.517394393692955508869058389691, −7.10604028804496855298145497397, −6.53747231897222211021137211021, −2.37714604467739077916333919542,
4.05533330791451947168279880511, 5.68070638546849464095536524868, 8.484102564279125782220274863224, 9.112705956177541863647570903426, 10.42251693501845675043344465453, 11.48138420354830170197930020217, 13.23246082696141745797793564410, 15.00864147381443500873456504151, 15.99776201174963910559487020643, 16.61092041346761771305940008744
