| L(s) = 1 | + (−1.42 − 0.687i)2-s + (−0.989 − 1.42i)3-s + (0.318 + 0.399i)4-s + (1.37 − 1.27i)5-s + (0.435 + 2.70i)6-s + (−2.57 − 1.48i)7-s + (0.525 + 2.30i)8-s + (−1.04 + 2.81i)9-s + (−2.84 + 0.876i)10-s + (−0.999 − 0.796i)11-s + (0.252 − 0.847i)12-s + (−4.01 − 1.23i)13-s + (2.65 + 3.88i)14-s + (−3.17 − 0.692i)15-s + (1.05 − 4.64i)16-s + (−2.30 + 2.48i)17-s + ⋯ |
| L(s) = 1 | + (−1.00 − 0.486i)2-s + (−0.571 − 0.820i)3-s + (0.159 + 0.199i)4-s + (0.615 − 0.571i)5-s + (0.177 + 1.10i)6-s + (−0.971 − 0.561i)7-s + (0.185 + 0.813i)8-s + (−0.346 + 0.937i)9-s + (−0.899 + 0.277i)10-s + (−0.301 − 0.240i)11-s + (0.0728 − 0.244i)12-s + (−1.11 − 0.343i)13-s + (0.708 + 1.03i)14-s + (−0.820 − 0.178i)15-s + (0.264 − 1.16i)16-s + (−0.559 + 0.603i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0154089 + 0.356463i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0154089 + 0.356463i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.989 + 1.42i)T \) |
| 43 | \( 1 + (5.81 + 3.03i)T \) |
| good | 2 | \( 1 + (1.42 + 0.687i)T + (1.24 + 1.56i)T^{2} \) |
| 5 | \( 1 + (-1.37 + 1.27i)T + (0.373 - 4.98i)T^{2} \) |
| 7 | \( 1 + (2.57 + 1.48i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.999 + 0.796i)T + (2.44 + 10.7i)T^{2} \) |
| 13 | \( 1 + (4.01 + 1.23i)T + (10.7 + 7.32i)T^{2} \) |
| 17 | \( 1 + (2.30 - 2.48i)T + (-1.27 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-3.18 - 1.25i)T + (13.9 + 12.9i)T^{2} \) |
| 23 | \( 1 + (-1.42 + 9.47i)T + (-21.9 - 6.77i)T^{2} \) |
| 29 | \( 1 + (-2.61 + 1.78i)T + (10.5 - 26.9i)T^{2} \) |
| 31 | \( 1 + (0.593 + 7.92i)T + (-30.6 + 4.62i)T^{2} \) |
| 37 | \( 1 + (-2.50 + 1.44i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.526 - 1.09i)T + (-25.5 - 32.0i)T^{2} \) |
| 47 | \( 1 + (-0.984 + 0.784i)T + (10.4 - 45.8i)T^{2} \) |
| 53 | \( 1 + (1.02 + 3.32i)T + (-43.7 + 29.8i)T^{2} \) |
| 59 | \( 1 + (2.32 + 0.530i)T + (53.1 + 25.5i)T^{2} \) |
| 61 | \( 1 + (-12.8 - 0.961i)T + (60.3 + 9.09i)T^{2} \) |
| 67 | \( 1 + (3.72 - 9.48i)T + (-49.1 - 45.5i)T^{2} \) |
| 71 | \( 1 + (5.19 - 0.782i)T + (67.8 - 20.9i)T^{2} \) |
| 73 | \( 1 + (-2.35 + 7.64i)T + (-60.3 - 41.1i)T^{2} \) |
| 79 | \( 1 + (0.705 - 1.22i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.28 + 7.74i)T + (-30.3 - 77.2i)T^{2} \) |
| 89 | \( 1 + (-8.40 - 5.72i)T + (32.5 + 82.8i)T^{2} \) |
| 97 | \( 1 + (-8.94 + 11.2i)T + (-21.5 - 94.5i)T^{2} \) |
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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
−12.83123593583784174685291113922, −11.68786438310494182355594998818, −10.46317876983365471351875101376, −9.875923391127255139519338042616, −8.674105215545204415998862097031, −7.54893090407700285798866838561, −6.24532350061717914519361477179, −5.01515859877643474067075959669, −2.34527595797542542772536541152, −0.53569197900663542617158425529,
3.13683936384439695010914054581, 5.04850991659797246919793240193, 6.40866422176197632328726840810, 7.26775457451735619911613358221, 9.031960381723728913296946152616, 9.652651380346655354963158337716, 10.22738574069309799544159383915, 11.61259836245608577672786901907, 12.72053230894497001322472703286, 13.98193653197623026035032861748
