| L(s) = 1 | + (−0.322 − 1.37i)2-s + (0.889 + 1.48i)3-s + (−1.79 + 0.887i)4-s + (0.479 + 0.174i)5-s + (1.76 − 1.70i)6-s + (1.63 + 0.287i)7-s + (1.79 + 2.18i)8-s + (−1.41 + 2.64i)9-s + (0.0858 − 0.716i)10-s + (−0.576 − 1.58i)11-s + (−2.91 − 1.87i)12-s + (3.81 + 4.54i)13-s + (−0.129 − 2.33i)14-s + (0.167 + 0.867i)15-s + (2.42 − 3.18i)16-s + (5.08 − 2.93i)17-s + ⋯ |
| L(s) = 1 | + (−0.227 − 0.973i)2-s + (0.513 + 0.858i)3-s + (−0.896 + 0.443i)4-s + (0.214 + 0.0780i)5-s + (0.718 − 0.695i)6-s + (0.616 + 0.108i)7-s + (0.636 + 0.771i)8-s + (−0.472 + 0.881i)9-s + (0.0271 − 0.226i)10-s + (−0.173 − 0.477i)11-s + (−0.840 − 0.541i)12-s + (1.05 + 1.26i)13-s + (−0.0346 − 0.625i)14-s + (0.0431 + 0.224i)15-s + (0.606 − 0.795i)16-s + (1.23 − 0.712i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0197i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0197i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.27043 - 0.0125306i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.27043 - 0.0125306i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.322 + 1.37i)T \) |
| 3 | \( 1 + (-0.889 - 1.48i)T \) |
| good | 5 | \( 1 + (-0.479 - 0.174i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-1.63 - 0.287i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (0.576 + 1.58i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-3.81 - 4.54i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-5.08 + 2.93i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.91 - 3.32i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.765 + 4.34i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (0.748 + 0.628i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (6.88 - 1.21i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-5.88 + 3.39i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.01 + 2.40i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.826 + 0.300i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.0117 + 0.0666i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 5.83T + 53T^{2} \) |
| 59 | \( 1 + (0.521 - 1.43i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-10.6 - 1.88i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (7.05 - 5.92i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (8.21 + 14.2i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.650 + 1.12i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.66 + 4.36i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (1.44 - 1.72i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-11.2 - 6.52i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-13.9 + 5.06i)T + (74.3 - 62.3i)T^{2} \) |
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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
−12.01716957555434015250712630319, −11.18981741084091865346557358966, −10.42699915984156564899850834798, −9.468255050681614725720453726304, −8.649629859535103883989979062690, −7.82636774576857448710040878156, −5.75390811815089617871522924811, −4.46228064733260955526202169513, −3.47971100840753171341355832765, −1.98575622707341581557308231067,
1.40273939972013847577273445814, 3.60557793422162159783207152220, 5.35853625081119014029519233752, 6.21796872070205573946738040469, 7.60217205369328911458158446416, 7.988107268561423145588578980482, 9.047383393555509387302423046554, 10.11327621489070543316731560695, 11.36034952163901363691091725758, 12.93237224790269548712834203841
