| L(s) = 1 | + 2i·3-s − 2i·7-s − 9-s + 2i·13-s − 6i·17-s + 4·19-s + 4·21-s + 6i·23-s + 4i·27-s + 6·29-s + 4·31-s − 2i·37-s − 4·39-s + 6·41-s + 10i·43-s + ⋯ |
| L(s) = 1 | + 1.15i·3-s − 0.755i·7-s − 0.333·9-s + 0.554i·13-s − 1.45i·17-s + 0.917·19-s + 0.872·21-s + 1.25i·23-s + 0.769i·27-s + 1.11·29-s + 0.718·31-s − 0.328i·37-s − 0.640·39-s + 0.937·41-s + 1.52i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.786291683\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.786291683\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2iT - 3T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 10iT - 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−9.513871003874312272869143754425, −9.187612383945413102225316721841, −7.85560562529560915898058281441, −7.26643077585279397498803879420, −6.29381265691880785284170621135, −5.09860777925640542831425659063, −4.60232248809496972242441447434, −3.69931065778313444397527750891, −2.82914192240295278569376546481, −1.10178150921557309047208423578,
0.881076883402113655854665800668, 2.05140123874588340287551303767, 2.93304773998191735352605200088, 4.22043832916332029336124542381, 5.37008833478037499539663023357, 6.19029348020805220188607267134, 6.76219782009384726421444939190, 7.79477033434019170830710090174, 8.286011623142564029466650767447, 9.051407543055912242328520920807
