| L(s) = 1 | + (−1.28 + 0.592i)2-s + (1.63 − 0.568i)3-s + (1.29 − 1.52i)4-s + (−3.25 − 2.49i)5-s + (−1.76 + 1.69i)6-s + (0.379 + 1.41i)7-s + (−0.764 + 2.72i)8-s + (2.35 − 1.85i)9-s + (5.66 + 1.27i)10-s + (0.325 + 0.0427i)11-s + (1.25 − 3.22i)12-s + (−0.713 − 5.41i)13-s + (−1.32 − 1.59i)14-s + (−6.74 − 2.23i)15-s + (−0.632 − 3.94i)16-s − 3.02i·17-s + ⋯ |
| L(s) = 1 | + (−0.907 + 0.419i)2-s + (0.944 − 0.327i)3-s + (0.648 − 0.760i)4-s + (−1.45 − 1.11i)5-s + (−0.720 + 0.693i)6-s + (0.143 + 0.535i)7-s + (−0.270 + 0.962i)8-s + (0.784 − 0.619i)9-s + (1.79 + 0.404i)10-s + (0.0979 + 0.0129i)11-s + (0.363 − 0.931i)12-s + (−0.197 − 1.50i)13-s + (−0.354 − 0.426i)14-s + (−1.74 − 0.578i)15-s + (−0.158 − 0.987i)16-s − 0.733i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.102 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.102 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.613439 - 0.553228i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.613439 - 0.553228i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.28 - 0.592i)T \) |
| 3 | \( 1 + (-1.63 + 0.568i)T \) |
| good | 5 | \( 1 + (3.25 + 2.49i)T + (1.29 + 4.82i)T^{2} \) |
| 7 | \( 1 + (-0.379 - 1.41i)T + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (-0.325 - 0.0427i)T + (10.6 + 2.84i)T^{2} \) |
| 13 | \( 1 + (0.713 + 5.41i)T + (-12.5 + 3.36i)T^{2} \) |
| 17 | \( 1 + 3.02iT - 17T^{2} \) |
| 19 | \( 1 + (2.69 + 1.11i)T + (13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (-0.526 + 1.96i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (4.59 + 5.98i)T + (-7.50 + 28.0i)T^{2} \) |
| 31 | \( 1 + (0.0686 + 0.118i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.03 - 1.25i)T + (26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (0.752 - 2.80i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (-6.69 - 0.881i)T + (41.5 + 11.1i)T^{2} \) |
| 47 | \( 1 + (-11.1 - 6.44i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.94 - 11.9i)T + (-37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (-5.70 - 4.38i)T + (15.2 + 56.9i)T^{2} \) |
| 61 | \( 1 + (-4.24 - 5.52i)T + (-15.7 + 58.9i)T^{2} \) |
| 67 | \( 1 + (7.77 - 1.02i)T + (64.7 - 17.3i)T^{2} \) |
| 71 | \( 1 + (-2.75 + 2.75i)T - 71iT^{2} \) |
| 73 | \( 1 + (-2.26 - 2.26i)T + 73iT^{2} \) |
| 79 | \( 1 + (13.1 + 7.58i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-6.84 + 5.25i)T + (21.4 - 80.1i)T^{2} \) |
| 89 | \( 1 + (1.07 - 1.07i)T - 89iT^{2} \) |
| 97 | \( 1 + (1.06 - 1.83i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.70568768695276487179807023900, −10.47193695974743710782030601778, −9.165359207967584282865871826416, −8.680101585014911774469945725626, −7.82321164744200455415901672654, −7.34738297124762768512271070472, −5.65860805048653604121451683633, −4.32439442036841180835234433888, −2.69846020491298608862542098473, −0.75662382691960909045436296899,
2.15163408003706762669200635085, 3.65329881485030191518755064810, 4.04303017611115695131181450229, 6.91381352139504186103427851331, 7.29026762349391972431973727988, 8.294041538508362671543399908338, 9.076169266476276458953696547645, 10.30750726013362543557154726190, 10.88479063187593523947533362081, 11.69518850736451016958079130440
