L(s) = 1 | − i·3-s + (2 − i)5-s + i·7-s + 2·9-s + 3·11-s − i·13-s + (−1 − 2i)15-s − 7i·17-s + 21-s + 6i·23-s + (3 − 4i)25-s − 5i·27-s − 5·29-s + 2·31-s − 3i·33-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + (0.894 − 0.447i)5-s + 0.377i·7-s + 0.666·9-s + 0.904·11-s − 0.277i·13-s + (−0.258 − 0.516i)15-s − 1.69i·17-s + 0.218·21-s + 1.25i·23-s + (0.600 − 0.800i)25-s − 0.962i·27-s − 0.928·29-s + 0.359·31-s − 0.522i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.417484863\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.417484863\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2 + i)T \) |
| 7 | \( 1 - iT \) |
good | 3 | \( 1 + iT - 3T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + iT - 13T^{2} \) |
| 17 | \( 1 + 7iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 3iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 10T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 - 5T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 7iT - 97T^{2} \) |
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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
−9.068412350255140306939146643709, −8.108186494296261588394792865077, −7.21818669667123939102654150910, −6.65936001273697007440744864994, −5.73196282873639369516574187779, −5.07989357237866836800762209516, −4.08071687170279223643863880464, −2.83810181262314492645977863271, −1.82791774459664028374294413967, −0.952524168864590401028320205886,
1.34036454047214452134835755629, 2.26999818288467770302564471837, 3.69018739425170334850656064369, 4.13101123223977134833991100736, 5.18586820535562694880024443271, 6.19052874324058501235793801000, 6.66901331191534618940759168829, 7.52269754290511291601646494203, 8.645238063030284943288355460034, 9.231343365479416086111233872857