| L(s) = 1 | + (−0.360 − 1.36i)2-s + (0.992 − 2.39i)3-s + (−1.74 + 0.985i)4-s + (0.0861 − 0.169i)5-s + (−3.63 − 0.493i)6-s + (−0.388 − 1.61i)7-s + (1.97 + 2.02i)8-s + (−2.63 − 2.63i)9-s + (−0.262 − 0.0568i)10-s + (−0.130 + 0.153i)11-s + (0.634 + 5.14i)12-s + (0.0548 + 0.0895i)13-s + (−2.07 + 1.11i)14-s + (−0.319 − 0.374i)15-s + (2.05 − 3.43i)16-s + (0.293 + 3.72i)17-s + ⋯ |
| L(s) = 1 | + (−0.254 − 0.966i)2-s + (0.572 − 1.38i)3-s + (−0.870 + 0.492i)4-s + (0.0385 − 0.0756i)5-s + (−1.48 − 0.201i)6-s + (−0.146 − 0.611i)7-s + (0.698 + 0.715i)8-s + (−0.877 − 0.877i)9-s + (−0.0829 − 0.0179i)10-s + (−0.0394 + 0.0461i)11-s + (0.183 + 1.48i)12-s + (0.0152 + 0.0248i)13-s + (−0.553 + 0.297i)14-s + (−0.0825 − 0.0966i)15-s + (0.514 − 0.857i)16-s + (0.0711 + 0.903i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.863 + 0.504i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.863 + 0.504i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.282523 - 1.04334i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.282523 - 1.04334i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.360 + 1.36i)T \) |
| 41 | \( 1 + (-3.90 + 5.07i)T \) |
| good | 3 | \( 1 + (-0.992 + 2.39i)T + (-2.12 - 2.12i)T^{2} \) |
| 5 | \( 1 + (-0.0861 + 0.169i)T + (-2.93 - 4.04i)T^{2} \) |
| 7 | \( 1 + (0.388 + 1.61i)T + (-6.23 + 3.17i)T^{2} \) |
| 11 | \( 1 + (0.130 - 0.153i)T + (-1.72 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.0548 - 0.0895i)T + (-5.90 + 11.5i)T^{2} \) |
| 17 | \( 1 + (-0.293 - 3.72i)T + (-16.7 + 2.65i)T^{2} \) |
| 19 | \( 1 + (-3.89 - 2.38i)T + (8.62 + 16.9i)T^{2} \) |
| 23 | \( 1 + (4.61 + 3.35i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.459 + 5.83i)T + (-28.6 - 4.53i)T^{2} \) |
| 31 | \( 1 + (-3.12 + 9.62i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-2.93 - 9.03i)T + (-29.9 + 21.7i)T^{2} \) |
| 43 | \( 1 + (-0.410 - 2.59i)T + (-40.8 + 13.2i)T^{2} \) |
| 47 | \( 1 + (2.73 - 11.3i)T + (-41.8 - 21.3i)T^{2} \) |
| 53 | \( 1 + (2.41 + 0.189i)T + (52.3 + 8.29i)T^{2} \) |
| 59 | \( 1 + (5.01 - 6.90i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (0.823 - 5.20i)T + (-58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (7.00 - 5.98i)T + (10.4 - 66.1i)T^{2} \) |
| 71 | \( 1 + (6.32 + 5.40i)T + (11.1 + 70.1i)T^{2} \) |
| 73 | \( 1 + (-5.47 + 5.47i)T - 73iT^{2} \) |
| 79 | \( 1 + (-4.74 - 1.96i)T + (55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 - 4.70iT - 83T^{2} \) |
| 89 | \( 1 + (5.68 - 1.36i)T + (79.2 - 40.4i)T^{2} \) |
| 97 | \( 1 + (10.4 - 8.89i)T + (15.1 - 95.8i)T^{2} \) |
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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
−12.46581301832045908119263221873, −11.68610283460128001478723260232, −10.43680726983330323868815448249, −9.422906294023353396315857867429, −8.120263544437003406782608301977, −7.63258254880060388415968817443, −6.14861939588687921310044384250, −4.14939429077905799809206847999, −2.66875893674098458445628346308, −1.24221463668343440622508017413,
3.20141603177071051412757331941, 4.64310967309098443846369661681, 5.53956936308937653153252686307, 7.03413879868884028330287315394, 8.372455952395430787428712418955, 9.194126692007365130296682828028, 9.837389495872169871058416832167, 10.84952730412552995056348072530, 12.34211694178964197577269267053, 13.84694386019723432230775971644
