L(s) = 1 | + 3.12i·3-s + (−1.32 + 1.80i)5-s − i·7-s − 6.76·9-s − 2.48·11-s − 4.15i·13-s + (−5.64 − 4.12i)15-s + 5.76i·17-s − 1.60·19-s + 3.12·21-s − 7.28i·23-s + (−1.51 − 4.76i)25-s − 11.7i·27-s + 1.45·29-s − 2.24·31-s + ⋯ |
L(s) = 1 | + 1.80i·3-s + (−0.590 + 0.807i)5-s − 0.377i·7-s − 2.25·9-s − 0.749·11-s − 1.15i·13-s + (−1.45 − 1.06i)15-s + 1.39i·17-s − 0.369·19-s + 0.681·21-s − 1.51i·23-s + (−0.303 − 0.952i)25-s − 2.26i·27-s + 0.270·29-s − 0.404·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.807 + 0.590i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.807 + 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3279753685\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3279753685\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.32 - 1.80i)T \) |
| 7 | \( 1 + iT \) |
good | 3 | \( 1 - 3.12iT - 3T^{2} \) |
| 11 | \( 1 + 2.48T + 11T^{2} \) |
| 13 | \( 1 + 4.15iT - 13T^{2} \) |
| 17 | \( 1 - 5.76iT - 17T^{2} \) |
| 19 | \( 1 + 1.60T + 19T^{2} \) |
| 23 | \( 1 + 7.28iT - 23T^{2} \) |
| 29 | \( 1 - 1.45T + 29T^{2} \) |
| 31 | \( 1 + 2.24T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 - 11.2T + 41T^{2} \) |
| 43 | \( 1 + 5.28iT - 43T^{2} \) |
| 47 | \( 1 - 3.45iT - 47T^{2} \) |
| 53 | \( 1 + 9.21iT - 53T^{2} \) |
| 59 | \( 1 + 5.92T + 59T^{2} \) |
| 61 | \( 1 + 5.35T + 61T^{2} \) |
| 67 | \( 1 - 7.52iT - 67T^{2} \) |
| 71 | \( 1 + 4.24T + 71T^{2} \) |
| 73 | \( 1 + 7.28iT - 73T^{2} \) |
| 79 | \( 1 + 16.9T + 79T^{2} \) |
| 83 | \( 1 + 10.1iT - 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 + 2.73iT - 97T^{2} \) |
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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
−8.957338954655511563705954915907, −8.259471302329889271609872358474, −7.68725955671088480477740942961, −6.40724163083226587627638702450, −5.71722021637013372551713533291, −4.69815035448418291999517402964, −4.10487345572472373686447857700, −3.29741240262584375005363382817, −2.60194272619800707753019422572, −0.12339308480590164766582464196,
1.11383796772185616752951366813, 2.08956047455687522791673897413, 3.00332364357698378627119875471, 4.36723065633882974752625138575, 5.36837312840749345534117531804, 6.01080203938411877728600457688, 7.08972625240239142253184506215, 7.50801132310055337453798675510, 8.097116736218823410479275305295, 9.067874294732085362845385729613