| L(s) = 1 | + (1.30 − 0.541i)2-s + (−1.15 + 5.82i)3-s + (1.41 − 1.41i)4-s + (−1.85 + 1.24i)5-s + (1.63 + 8.24i)6-s + (8.10 + 5.41i)7-s + (1.08 − 2.61i)8-s + (−24.3 − 10.0i)9-s + (−1.75 + 2.62i)10-s + (−8.63 + 1.71i)11-s + (6.60 + 9.88i)12-s + (−2.55 − 2.55i)13-s + (13.5 + 2.68i)14-s + (−5.08 − 12.2i)15-s − 4i·16-s + (1.16 + 16.9i)17-s + ⋯ |
| L(s) = 1 | + (0.653 − 0.270i)2-s + (−0.386 + 1.94i)3-s + (0.353 − 0.353i)4-s + (−0.371 + 0.248i)5-s + (0.273 + 1.37i)6-s + (1.15 + 0.773i)7-s + (0.135 − 0.326i)8-s + (−2.70 − 1.11i)9-s + (−0.175 + 0.262i)10-s + (−0.784 + 0.156i)11-s + (0.550 + 0.823i)12-s + (−0.196 − 0.196i)13-s + (0.965 + 0.192i)14-s + (−0.339 − 0.818i)15-s − 0.250i·16-s + (0.0685 + 0.997i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.495 - 0.868i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.495 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.861761 + 1.48290i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.861761 + 1.48290i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.30 + 0.541i)T \) |
| 5 | \( 1 + (1.85 - 1.24i)T \) |
| 17 | \( 1 + (-1.16 - 16.9i)T \) |
| good | 3 | \( 1 + (1.15 - 5.82i)T + (-8.31 - 3.44i)T^{2} \) |
| 7 | \( 1 + (-8.10 - 5.41i)T + (18.7 + 45.2i)T^{2} \) |
| 11 | \( 1 + (8.63 - 1.71i)T + (111. - 46.3i)T^{2} \) |
| 13 | \( 1 + (2.55 + 2.55i)T + 169iT^{2} \) |
| 19 | \( 1 + (-18.9 + 7.85i)T + (255. - 255. i)T^{2} \) |
| 23 | \( 1 + (-4.99 - 25.1i)T + (-488. + 202. i)T^{2} \) |
| 29 | \( 1 + (-8.21 - 12.2i)T + (-321. + 776. i)T^{2} \) |
| 31 | \( 1 + (7.13 + 1.41i)T + (887. + 367. i)T^{2} \) |
| 37 | \( 1 + (-9.60 + 48.2i)T + (-1.26e3 - 523. i)T^{2} \) |
| 41 | \( 1 + (-58.1 - 38.8i)T + (643. + 1.55e3i)T^{2} \) |
| 43 | \( 1 + (-50.0 - 20.7i)T + (1.30e3 + 1.30e3i)T^{2} \) |
| 47 | \( 1 + (4.80 + 4.80i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (11.1 - 4.62i)T + (1.98e3 - 1.98e3i)T^{2} \) |
| 59 | \( 1 + (-30.0 + 72.5i)T + (-2.46e3 - 2.46e3i)T^{2} \) |
| 61 | \( 1 + (-30.8 + 46.2i)T + (-1.42e3 - 3.43e3i)T^{2} \) |
| 67 | \( 1 - 26.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (4.78 - 24.0i)T + (-4.65e3 - 1.92e3i)T^{2} \) |
| 73 | \( 1 + (49.2 - 32.9i)T + (2.03e3 - 4.92e3i)T^{2} \) |
| 79 | \( 1 + (99.7 - 19.8i)T + (5.76e3 - 2.38e3i)T^{2} \) |
| 83 | \( 1 + (17.2 + 41.6i)T + (-4.87e3 + 4.87e3i)T^{2} \) |
| 89 | \( 1 + (7.83 - 7.83i)T - 7.92e3iT^{2} \) |
| 97 | \( 1 + (58.7 + 87.8i)T + (-3.60e3 + 8.69e3i)T^{2} \) |
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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
−12.62055101054884077422158942823, −11.37064978838249339541352879726, −11.17714634624547873182821250697, −10.11283057963919708094785318764, −9.104895610048744659440066050945, −7.87891848276092071626448953981, −5.73321036350283781468643014673, −5.13738854222354165272981407323, −4.12008260838297434575881161885, −2.85762531862082580603636095618,
0.943274241815640542577418234797, 2.58285827993466673584559943613, 4.76771522940388353165244022224, 5.78514365332450703968615125964, 7.19754013552925125124722447518, 7.59864499514395879804671503002, 8.482341436203618284985628259845, 10.79844855744218989519312124076, 11.62846850549614841658364897671, 12.25970515361276816645852612625
