L(s) = 1 | + (−1.22 + 2.12i)5-s + (0.5 + 2.59i)7-s + (−3.67 + 2.12i)11-s − 1.73i·13-s + (2.44 + 4.24i)17-s + (4.5 + 2.59i)19-s + (−0.499 − 0.866i)25-s − 8.48i·29-s + (−1.5 + 0.866i)31-s + (−6.12 − 2.12i)35-s + (2.5 − 4.33i)37-s + 12.2·41-s − 11·43-s + (−1.22 + 2.12i)47-s + (−6.5 + 2.59i)49-s + ⋯ |
L(s) = 1 | + (−0.547 + 0.948i)5-s + (0.188 + 0.981i)7-s + (−1.10 + 0.639i)11-s − 0.480i·13-s + (0.594 + 1.02i)17-s + (1.03 + 0.596i)19-s + (−0.0999 − 0.173i)25-s − 1.57i·29-s + (−0.269 + 0.155i)31-s + (−1.03 − 0.358i)35-s + (0.410 − 0.711i)37-s + 1.91·41-s − 1.67·43-s + (−0.178 + 0.309i)47-s + (−0.928 + 0.371i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0285 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0285 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.741913 + 0.721060i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.741913 + 0.721060i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.5 - 2.59i)T \) |
good | 5 | \( 1 + (1.22 - 2.12i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (3.67 - 2.12i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 1.73iT - 13T^{2} \) |
| 17 | \( 1 + (-2.44 - 4.24i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.5 - 2.59i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 8.48iT - 29T^{2} \) |
| 31 | \( 1 + (1.5 - 0.866i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.5 + 4.33i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 12.2T + 41T^{2} \) |
| 43 | \( 1 + 11T + 43T^{2} \) |
| 47 | \( 1 + (1.22 - 2.12i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-7.34 + 4.24i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2.44 + 4.24i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3 + 1.73i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.5 - 6.06i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 12.7iT - 71T^{2} \) |
| 73 | \( 1 + (-13.5 + 7.79i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.5 - 9.52i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 12.2T + 83T^{2} \) |
| 89 | \( 1 + (7.34 - 12.7i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 3.46iT - 97T^{2} \) |
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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
−12.23665311727389713436249316183, −11.32495128941803388241879613500, −10.42474489861764702556752935630, −9.557149975596285705003621283940, −8.050154204324423828593500120443, −7.62108095900743968520313492496, −6.16334873165294198695912717051, −5.19425963867059200122406711247, −3.56719629838045156224940560503, −2.37366943079236095456801294297,
0.858704779014231631331359456110, 3.16853055523483917202613613921, 4.57474437931164353483963472192, 5.36974936038712407766431862445, 7.08696045026379803266787478060, 7.84766420295975051561484822185, 8.826132711472494254159353082158, 9.892313649185229154948720974856, 10.97922009435136136279441009713, 11.74901254245007559201780991242