| L(s) = 1 | − 1.23i·3-s + 3.23i·7-s + 1.47·9-s + 2·11-s + 4.47i·13-s − 4.47i·17-s − 4.47·19-s + 4.00·21-s − 4.76i·23-s − 5.52i·27-s − 2·29-s + 6.47·31-s − 2.47i·33-s + 6.94i·37-s + 5.52·39-s + ⋯ |
| L(s) = 1 | − 0.713i·3-s + 1.22i·7-s + 0.490·9-s + 0.603·11-s + 1.24i·13-s − 1.08i·17-s − 1.02·19-s + 0.872·21-s − 0.993i·23-s − 1.06i·27-s − 0.371·29-s + 1.16·31-s − 0.430i·33-s + 1.14i·37-s + 0.885·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.930260060\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.930260060\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 1.23iT - 3T^{2} \) |
| 7 | \( 1 - 3.23iT - 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 4.47iT - 13T^{2} \) |
| 17 | \( 1 + 4.47iT - 17T^{2} \) |
| 19 | \( 1 + 4.47T + 19T^{2} \) |
| 23 | \( 1 + 4.76iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 6.47T + 31T^{2} \) |
| 37 | \( 1 - 6.94iT - 37T^{2} \) |
| 41 | \( 1 - 12.4T + 41T^{2} \) |
| 43 | \( 1 - 7.70iT - 43T^{2} \) |
| 47 | \( 1 - 7.23iT - 47T^{2} \) |
| 53 | \( 1 + 0.472iT - 53T^{2} \) |
| 59 | \( 1 + 8.47T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 7.70iT - 67T^{2} \) |
| 71 | \( 1 + 2.47T + 71T^{2} \) |
| 73 | \( 1 - 4.47iT - 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 - 3.70iT - 83T^{2} \) |
| 89 | \( 1 - 14.9T + 89T^{2} \) |
| 97 | \( 1 - 16.4iT - 97T^{2} \) |
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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
−8.750647562608761906277237728090, −7.994669380261486652202558266358, −7.11475175259393129499144862821, −6.42117225820839202743109621410, −6.03861256087492843310973712619, −4.69185978017375178835822489298, −4.28677012319429368409776922070, −2.77474420935656507470500357144, −2.16656593345235083930691972699, −1.09900904326898180862302296479,
0.69508389352514619084133947342, 1.86484696069968941858677198884, 3.34704632588316284175840645526, 3.97792781329446636821667813789, 4.47494439584089825958156170670, 5.55833087780787223208604571985, 6.30953712883732295337473549453, 7.27010876747113059594623467309, 7.75257268658025298980962681865, 8.683783781265074346666273999811
