| L(s) = 1 | + (−1.36 − 0.119i)2-s + (−0.861 − 1.50i)3-s + (−0.116 − 0.0204i)4-s + (−0.119 + 2.23i)5-s + (0.997 + 2.15i)6-s + (−2.68 − 1.87i)7-s + (2.80 + 0.752i)8-s + (−1.51 + 2.58i)9-s + (0.430 − 3.03i)10-s + (−1.85 + 5.10i)11-s + (0.0692 + 0.192i)12-s + (0.215 + 2.45i)13-s + (3.44 + 2.88i)14-s + (3.45 − 1.74i)15-s + (−3.52 − 1.28i)16-s + (−3.45 + 0.925i)17-s + ⋯ |
| L(s) = 1 | + (−0.966 − 0.0845i)2-s + (−0.497 − 0.867i)3-s + (−0.0580 − 0.0102i)4-s + (−0.0533 + 0.998i)5-s + (0.407 + 0.880i)6-s + (−1.01 − 0.710i)7-s + (0.992 + 0.265i)8-s + (−0.505 + 0.862i)9-s + (0.136 − 0.960i)10-s + (−0.560 + 1.53i)11-s + (0.0199 + 0.0554i)12-s + (0.0596 + 0.682i)13-s + (0.920 + 0.772i)14-s + (0.892 − 0.450i)15-s + (−0.881 − 0.320i)16-s + (−0.837 + 0.224i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.469 - 0.882i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.469 - 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0846385 + 0.140863i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0846385 + 0.140863i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.861 + 1.50i)T \) |
| 5 | \( 1 + (0.119 - 2.23i)T \) |
| good | 2 | \( 1 + (1.36 + 0.119i)T + (1.96 + 0.347i)T^{2} \) |
| 7 | \( 1 + (2.68 + 1.87i)T + (2.39 + 6.57i)T^{2} \) |
| 11 | \( 1 + (1.85 - 5.10i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.215 - 2.45i)T + (-12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (3.45 - 0.925i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-0.417 + 0.240i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.01 + 5.73i)T + (-7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (-0.0993 + 0.0833i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.509 + 2.88i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-1.67 - 6.26i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-0.215 + 0.256i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (2.45 + 5.27i)T + (-27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (6.80 - 9.71i)T + (-16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (0.167 - 0.167i)T - 53iT^{2} \) |
| 59 | \( 1 + (3.62 - 1.32i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (0.855 + 4.85i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-4.25 + 0.372i)T + (65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (-7.10 - 4.10i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (3.51 - 13.1i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-6.44 - 7.68i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-0.00128 + 0.0147i)T + (-81.7 - 14.4i)T^{2} \) |
| 89 | \( 1 + (2.55 + 4.43i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.77 + 1.75i)T + (62.3 - 74.3i)T^{2} \) |
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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
−13.44748554850027269906139242784, −12.59413874793310302560569537325, −11.26775661047262932431795901118, −10.32526251524121876256534675736, −9.710986843617434906882705433084, −8.105811161854005635751083915167, −7.10922717710888462637754563913, −6.49575125825363980911639074562, −4.45842657801523749954061518461, −2.20015788709963405912677552833,
0.23588619837239957452139932576, 3.53136545399095718038677124698, 5.12019090023020896473262555264, 6.08443830076744793438663787040, 8.067418668961352930522505155499, 8.933685779377101258502196756266, 9.552395813392142163592891988656, 10.56993201646825437912197519755, 11.66776345488512927159758560385, 12.89951391028696613386019871847
