| L(s) = 1 | + (1.06 + 0.928i)2-s + (0.275 + 1.98i)4-s + (−0.605 + 1.66i)5-s + (0.748 + 0.131i)7-s + (−1.54 + 2.36i)8-s + (−2.18 + 1.21i)10-s + (3.01 − 1.09i)11-s + (−1.07 + 0.899i)13-s + (0.675 + 0.835i)14-s + (−3.84 + 1.09i)16-s + (−5.55 + 3.20i)17-s + (2.51 + 1.45i)19-s + (−3.45 − 0.740i)20-s + (4.23 + 1.63i)22-s + (−1.06 − 6.06i)23-s + ⋯ |
| L(s) = 1 | + (0.754 + 0.656i)2-s + (0.137 + 0.990i)4-s + (−0.270 + 0.743i)5-s + (0.282 + 0.0498i)7-s + (−0.546 + 0.837i)8-s + (−0.692 + 0.383i)10-s + (0.909 − 0.331i)11-s + (−0.297 + 0.249i)13-s + (0.180 + 0.223i)14-s + (−0.962 + 0.272i)16-s + (−1.34 + 0.778i)17-s + (0.577 + 0.333i)19-s + (−0.773 − 0.165i)20-s + (0.903 + 0.347i)22-s + (−0.222 − 1.26i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.235 - 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.235 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.15744 + 1.47136i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.15744 + 1.47136i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.06 - 0.928i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (0.605 - 1.66i)T + (-3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.748 - 0.131i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-3.01 + 1.09i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (1.07 - 0.899i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (5.55 - 3.20i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.51 - 1.45i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.06 + 6.06i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-4.87 + 5.80i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-9.30 + 1.64i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-1.62 - 2.80i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.14 - 4.94i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (2.50 + 6.87i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.737 + 4.18i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 9.63iT - 53T^{2} \) |
| 59 | \( 1 + (-0.397 - 0.144i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.38 + 7.85i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (3.65 + 4.35i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.88 - 3.27i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (5.59 - 9.69i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (6.53 - 7.78i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-3.80 - 3.19i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-0.509 - 0.294i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-15.8 + 5.78i)T + (74.3 - 62.3i)T^{2} \) |
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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
−11.79020036176236601633960960958, −11.34653598698177402155906856293, −10.11348515552031466205314485122, −8.715007962418485601359217100063, −8.004161128352334525100660507183, −6.66708056246388563736295510637, −6.36220385939079908898975051341, −4.76756580920850596560751904578, −3.85758261964524107394373333941, −2.52008177960518395435357553552,
1.21579830054525790332906134257, 2.87089697875004293672240042241, 4.37780148853100319342563746837, 4.91403986558812374129109255116, 6.28068775193034337952865854820, 7.36066727193364145390584402380, 8.841241416497857274726633968606, 9.498415563881427354526124611302, 10.64466684578468737422550096533, 11.66037367532111286041060351814
