L(s) = 1 | − 0.496i·3-s + (0.987 − 2.00i)5-s + (1.55 + 1.55i)7-s + 2.75·9-s + (4.19 + 4.19i)11-s − 5.09·13-s + (−0.996 − 0.490i)15-s + (0.213 + 0.213i)17-s + (0.844 + 0.844i)19-s + (0.771 − 0.771i)21-s + (1.70 − 1.70i)23-s + (−3.05 − 3.96i)25-s − 2.85i·27-s + (2.24 − 2.24i)29-s − 0.818i·31-s + ⋯ |
L(s) = 1 | − 0.286i·3-s + (0.441 − 0.897i)5-s + (0.587 + 0.587i)7-s + 0.917·9-s + (1.26 + 1.26i)11-s − 1.41·13-s + (−0.257 − 0.126i)15-s + (0.0517 + 0.0517i)17-s + (0.193 + 0.193i)19-s + (0.168 − 0.168i)21-s + (0.356 − 0.356i)23-s + (−0.610 − 0.792i)25-s − 0.549i·27-s + (0.417 − 0.417i)29-s − 0.146i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 + 0.336i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.941 + 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.82577 - 0.316350i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.82577 - 0.316350i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.987 + 2.00i)T \) |
good | 3 | \( 1 + 0.496iT - 3T^{2} \) |
| 7 | \( 1 + (-1.55 - 1.55i)T + 7iT^{2} \) |
| 11 | \( 1 + (-4.19 - 4.19i)T + 11iT^{2} \) |
| 13 | \( 1 + 5.09T + 13T^{2} \) |
| 17 | \( 1 + (-0.213 - 0.213i)T + 17iT^{2} \) |
| 19 | \( 1 + (-0.844 - 0.844i)T + 19iT^{2} \) |
| 23 | \( 1 + (-1.70 + 1.70i)T - 23iT^{2} \) |
| 29 | \( 1 + (-2.24 + 2.24i)T - 29iT^{2} \) |
| 31 | \( 1 + 0.818iT - 31T^{2} \) |
| 37 | \( 1 - 5.12T + 37T^{2} \) |
| 41 | \( 1 + 3.34iT - 41T^{2} \) |
| 43 | \( 1 - 4.49T + 43T^{2} \) |
| 47 | \( 1 + (4.29 - 4.29i)T - 47iT^{2} \) |
| 53 | \( 1 + 1.00iT - 53T^{2} \) |
| 59 | \( 1 + (7.65 - 7.65i)T - 59iT^{2} \) |
| 61 | \( 1 + (-1.90 - 1.90i)T + 61iT^{2} \) |
| 67 | \( 1 + 11.0T + 67T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 + (2.70 + 2.70i)T + 73iT^{2} \) |
| 79 | \( 1 - 8.32T + 79T^{2} \) |
| 83 | \( 1 + 9.17iT - 83T^{2} \) |
| 89 | \( 1 - 4.25T + 89T^{2} \) |
| 97 | \( 1 + (7.15 + 7.15i)T + 97iT^{2} \) |
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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
−10.25011170257110809511525817812, −9.574663430194101798280469222166, −8.984333916461815862960503167591, −7.80436328213940018101553672359, −7.08066000467168490847619462767, −6.00884040988074505201987472740, −4.77038950399597520990137026442, −4.36959655178149839104502521326, −2.29467225643245819847368826571, −1.38705872057084113487625641798,
1.37938237224319441576763492955, 2.94374327745571939394010834105, 4.02094281308057997090199392539, 5.03141752038553053863029966775, 6.27408982760367688392778045267, 7.06468564607695466453596586572, 7.80278273613473692910089460454, 9.151635363049894957109500446329, 9.775352684127258804298953553279, 10.64803157722379391651096396759