| L(s) = 1 | + (0.580 + 1.91i)2-s + (−2.34 − 1.87i)3-s + (−3.32 + 2.22i)4-s + (0.130 + 0.741i)5-s + (2.22 − 5.57i)6-s + (−1.32 − 1.11i)7-s + (−6.18 − 5.07i)8-s + (1.97 + 8.78i)9-s + (−1.34 + 0.681i)10-s + (2.40 − 13.6i)11-s + (11.9 + 1.02i)12-s + (5.13 − 14.1i)13-s + (1.35 − 3.17i)14-s + (1.08 − 1.98i)15-s + (6.11 − 14.7i)16-s + (7.61 − 4.39i)17-s + ⋯ |
| L(s) = 1 | + (0.290 + 0.956i)2-s + (−0.780 − 0.624i)3-s + (−0.831 + 0.555i)4-s + (0.0261 + 0.148i)5-s + (0.370 − 0.928i)6-s + (−0.189 − 0.158i)7-s + (−0.773 − 0.633i)8-s + (0.219 + 0.975i)9-s + (−0.134 + 0.0681i)10-s + (0.218 − 1.24i)11-s + (0.996 + 0.0851i)12-s + (0.395 − 1.08i)13-s + (0.0969 − 0.227i)14-s + (0.0722 − 0.132i)15-s + (0.381 − 0.924i)16-s + (0.448 − 0.258i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.660 + 0.750i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.660 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.827181 - 0.373916i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.827181 - 0.373916i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.580 - 1.91i)T \) |
| 3 | \( 1 + (2.34 + 1.87i)T \) |
| good | 5 | \( 1 + (-0.130 - 0.741i)T + (-23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (1.32 + 1.11i)T + (8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (-2.40 + 13.6i)T + (-113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (-5.13 + 14.1i)T + (-129. - 108. i)T^{2} \) |
| 17 | \( 1 + (-7.61 + 4.39i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (7.40 + 4.27i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (16.0 + 19.0i)T + (-91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (31.3 - 11.4i)T + (644. - 540. i)T^{2} \) |
| 31 | \( 1 + (-33.4 + 28.0i)T + (166. - 946. i)T^{2} \) |
| 37 | \( 1 + (26.0 - 15.0i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (16.2 - 44.7i)T + (-1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (24.9 + 4.39i)T + (1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (31.4 - 37.4i)T + (-383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 - 16.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + (6.61 + 37.5i)T + (-3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (39.1 - 46.6i)T + (-646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-40.7 + 112. i)T + (-3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (36.2 - 20.9i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-52.5 + 91.0i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-31.0 + 11.3i)T + (4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (70.8 - 25.7i)T + (5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (4.93 + 2.85i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-14.9 + 84.7i)T + (-8.84e3 - 3.21e3i)T^{2} \) |
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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
−12.18335405151683925677409990268, −11.11259357375062359845004014667, −10.07758758251757328263190486533, −8.532341340165875646475394950526, −7.83254369577516286054881002383, −6.56755056580012768168144999911, −5.98006767311537659406993609443, −4.87225319309681515601914968113, −3.25060381156400461955952193276, −0.52862813936605131749675070800,
1.67087101616576619749282736673, 3.66130068132427144827699867995, 4.58653417113080049249038520042, 5.65301284909996411428080889301, 6.85894375462505815167962790577, 8.756652475070604939115557891843, 9.645027144889260266569553259035, 10.31000877647821635914016840186, 11.38015241741562793572466829909, 12.09737953402518671265364216262
