L(s) = 1 | + (0.438 − 1.95i)2-s − 1.73i·3-s + (−3.61 − 1.71i)4-s + 2.23·5-s + (−3.37 − 0.758i)6-s − 6.33i·7-s + (−4.92 + 6.30i)8-s − 2.99·9-s + (0.979 − 4.36i)10-s + 9.27i·11-s + (−2.96 + 6.26i)12-s + 18.5·13-s + (−12.3 − 2.77i)14-s − 3.87i·15-s + (10.1 + 12.3i)16-s + 13.9·17-s + ⋯ |
L(s) = 1 | + (0.219 − 0.975i)2-s − 0.577i·3-s + (−0.904 − 0.427i)4-s + 0.447·5-s + (−0.563 − 0.126i)6-s − 0.904i·7-s + (−0.615 + 0.788i)8-s − 0.333·9-s + (0.0979 − 0.436i)10-s + 0.843i·11-s + (−0.246 + 0.521i)12-s + 1.42·13-s + (−0.882 − 0.198i)14-s − 0.258i·15-s + (0.634 + 0.772i)16-s + 0.818·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.427 + 0.904i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.427 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.681091 - 1.07552i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.681091 - 1.07552i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.438 + 1.95i)T \) |
| 3 | \( 1 + 1.73iT \) |
| 5 | \( 1 - 2.23T \) |
good | 7 | \( 1 + 6.33iT - 49T^{2} \) |
| 11 | \( 1 - 9.27iT - 121T^{2} \) |
| 13 | \( 1 - 18.5T + 169T^{2} \) |
| 17 | \( 1 - 13.9T + 289T^{2} \) |
| 19 | \( 1 + 17.2iT - 361T^{2} \) |
| 23 | \( 1 - 33.7iT - 529T^{2} \) |
| 29 | \( 1 + 28.6T + 841T^{2} \) |
| 31 | \( 1 - 23.4iT - 961T^{2} \) |
| 37 | \( 1 + 67.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + 44.0T + 1.68e3T^{2} \) |
| 43 | \( 1 + 50.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 31.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 81.6T + 2.80e3T^{2} \) |
| 59 | \( 1 - 19.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 53.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 4.49iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 13.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 40.8T + 5.32e3T^{2} \) |
| 79 | \( 1 - 141. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 69.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 46.3T + 7.92e3T^{2} \) |
| 97 | \( 1 - 68.5T + 9.40e3T^{2} \) |
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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
−13.84348089718287515247501491800, −13.47872910896374049771341996243, −12.29527908748590497283438806688, −11.10268012491640816014717248828, −10.09297215465649447814753484238, −8.805488673974056324019446471669, −7.15178378857471529654629147460, −5.44242165832381185361714112609, −3.62191835015240483968065384459, −1.48753735575918638957670754400,
3.57812825554917118722437595741, 5.44038355124807949876542633544, 6.24057895797071930214489246274, 8.265472651007401278552177569076, 9.020735408701690639123978765274, 10.41809610707503167198972507060, 11.99993788810681609870392801529, 13.30449264297834966319924241178, 14.30090505194040021162333962655, 15.23852027462976070997756152882