| L(s) = 1 | + (−0.686 − 0.727i)2-s + (0.862 + 1.50i)3-s + (−0.0581 + 0.998i)4-s + (−3.71 + 0.434i)5-s + (0.500 − 1.65i)6-s + (3.11 + 1.56i)7-s + (0.766 − 0.642i)8-s + (−1.51 + 2.59i)9-s + (2.86 + 2.40i)10-s + (1.94 + 4.51i)11-s + (−1.54 + 0.773i)12-s + (−2.77 − 0.657i)13-s + (−0.999 − 3.33i)14-s + (−3.85 − 5.20i)15-s + (−0.993 − 0.116i)16-s + (−0.261 + 1.48i)17-s + ⋯ |
| L(s) = 1 | + (−0.485 − 0.514i)2-s + (0.498 + 0.867i)3-s + (−0.0290 + 0.499i)4-s + (−1.66 + 0.194i)5-s + (0.204 − 0.676i)6-s + (1.17 + 0.591i)7-s + (0.270 − 0.227i)8-s + (−0.503 + 0.863i)9-s + (0.906 + 0.760i)10-s + (0.587 + 1.36i)11-s + (−0.447 + 0.223i)12-s + (−0.769 − 0.182i)13-s + (−0.267 − 0.892i)14-s + (−0.996 − 1.34i)15-s + (−0.248 − 0.0290i)16-s + (−0.0634 + 0.359i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.332 - 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.332 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.687336 + 0.486593i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.687336 + 0.486593i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.686 + 0.727i)T \) |
| 3 | \( 1 + (-0.862 - 1.50i)T \) |
| good | 5 | \( 1 + (3.71 - 0.434i)T + (4.86 - 1.15i)T^{2} \) |
| 7 | \( 1 + (-3.11 - 1.56i)T + (4.18 + 5.61i)T^{2} \) |
| 11 | \( 1 + (-1.94 - 4.51i)T + (-7.54 + 8.00i)T^{2} \) |
| 13 | \( 1 + (2.77 + 0.657i)T + (11.6 + 5.83i)T^{2} \) |
| 17 | \( 1 + (0.261 - 1.48i)T + (-15.9 - 5.81i)T^{2} \) |
| 19 | \( 1 + (0.0992 + 0.563i)T + (-17.8 + 6.49i)T^{2} \) |
| 23 | \( 1 + (-5.45 + 2.73i)T + (13.7 - 18.4i)T^{2} \) |
| 29 | \( 1 + (-2.50 + 8.36i)T + (-24.2 - 15.9i)T^{2} \) |
| 31 | \( 1 + (1.77 + 1.16i)T + (12.2 + 28.4i)T^{2} \) |
| 37 | \( 1 + (-3.89 + 1.41i)T + (28.3 - 23.7i)T^{2} \) |
| 41 | \( 1 + (-0.955 + 1.01i)T + (-2.38 - 40.9i)T^{2} \) |
| 43 | \( 1 + (2.05 - 2.75i)T + (-12.3 - 41.1i)T^{2} \) |
| 47 | \( 1 + (-4.01 + 2.63i)T + (18.6 - 43.1i)T^{2} \) |
| 53 | \( 1 + (3.41 - 5.91i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.76 + 4.10i)T + (-40.4 - 42.9i)T^{2} \) |
| 61 | \( 1 + (-0.555 - 9.53i)T + (-60.5 + 7.08i)T^{2} \) |
| 67 | \( 1 + (-1.84 - 6.17i)T + (-55.9 + 36.8i)T^{2} \) |
| 71 | \( 1 + (-12.4 - 10.4i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (-4.77 + 4.00i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (11.0 + 11.6i)T + (-4.59 + 78.8i)T^{2} \) |
| 83 | \( 1 + (-1.00 - 1.06i)T + (-4.82 + 82.8i)T^{2} \) |
| 89 | \( 1 + (3.39 - 2.84i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-9.47 - 1.10i)T + (94.3 + 22.3i)T^{2} \) |
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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
−12.64947537623224975131163051270, −11.73518520620528992724024927323, −11.19084773643680689979172454165, −10.06794967711454089363669512755, −8.937147512068571511619507750376, −8.072966584062178777857778220120, −7.32723524682015433012518347454, −4.77127109172696390552546532467, −4.10879037407184474921819758127, −2.49711260033050850401523044834,
0.982492822554157887070919041992, 3.47807418872303349428991798637, 4.95889364808430003764707759411, 6.82223411899364062499292976646, 7.59764860217271881569774091699, 8.289227035986253368155702305086, 9.055963648881159869769145426812, 11.06194537828427031210065313996, 11.49589754342891909626367173264, 12.58794924493520214008134498592
