| L(s) = 1 | + (−1.37 + 0.340i)2-s + (2.65 + 1.81i)3-s + (1.76 − 0.934i)4-s + (−3.39 − 0.254i)5-s + (−4.26 − 1.58i)6-s + (0.227 + 2.63i)7-s + (−2.10 + 1.88i)8-s + (2.68 + 6.83i)9-s + (4.74 − 0.806i)10-s + (−0.627 − 0.246i)11-s + (6.39 + 0.719i)12-s + (0.189 − 0.151i)13-s + (−1.20 − 3.54i)14-s + (−8.56 − 6.82i)15-s + (2.25 − 3.30i)16-s + (2.64 + 2.84i)17-s + ⋯ |
| L(s) = 1 | + (−0.970 + 0.240i)2-s + (1.53 + 1.04i)3-s + (0.884 − 0.467i)4-s + (−1.51 − 0.113i)5-s + (−1.74 − 0.645i)6-s + (0.0858 + 0.996i)7-s + (−0.745 + 0.666i)8-s + (0.894 + 2.27i)9-s + (1.50 − 0.255i)10-s + (−0.189 − 0.0742i)11-s + (1.84 + 0.207i)12-s + (0.0525 − 0.0419i)13-s + (−0.323 − 0.946i)14-s + (−2.21 − 1.76i)15-s + (0.563 − 0.826i)16-s + (0.641 + 0.691i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.186 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.186 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.641027 + 0.774475i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.641027 + 0.774475i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.37 - 0.340i)T \) |
| 7 | \( 1 + (-0.227 - 2.63i)T \) |
| good | 3 | \( 1 + (-2.65 - 1.81i)T + (1.09 + 2.79i)T^{2} \) |
| 5 | \( 1 + (3.39 + 0.254i)T + (4.94 + 0.745i)T^{2} \) |
| 11 | \( 1 + (0.627 + 0.246i)T + (8.06 + 7.48i)T^{2} \) |
| 13 | \( 1 + (-0.189 + 0.151i)T + (2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (-2.64 - 2.84i)T + (-1.27 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-1.77 + 3.07i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.03 + 3.27i)T + (-1.71 - 22.9i)T^{2} \) |
| 29 | \( 1 + (0.473 + 2.07i)T + (-26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (-0.307 - 0.532i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.57 + 0.485i)T + (30.5 + 20.8i)T^{2} \) |
| 41 | \( 1 + (1.74 + 3.61i)T + (-25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (-2.65 + 5.50i)T + (-26.8 - 33.6i)T^{2} \) |
| 47 | \( 1 + (-0.966 + 0.145i)T + (44.9 - 13.8i)T^{2} \) |
| 53 | \( 1 + (-8.02 + 2.47i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (0.668 + 8.92i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (1.03 - 3.34i)T + (-50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (-10.4 + 6.03i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-9.25 - 2.11i)T + (63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (2.30 - 15.3i)T + (-69.7 - 21.5i)T^{2} \) |
| 79 | \( 1 + (8.66 + 5.00i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (2.81 - 3.53i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (2.77 - 1.08i)T + (65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 - 0.505iT - 97T^{2} \) |
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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
−12.61531666973513792167073376473, −11.49083263486802013568825843948, −10.56536172163568744169804851446, −9.502129699672768707789404043699, −8.565433014983015847210336391562, −8.251012712643700087015518043670, −7.22690788204821036857675180447, −5.16823456062782357440200005853, −3.70750714701103823229869750166, −2.57270604777212526574243901118,
1.13445068121144291336095562312, 3.02339769872739815050535782468, 3.82445548822777585756374273292, 6.88029657804592882718544761341, 7.59310774088054569961109292534, 7.911138547107778871034182968104, 8.960547150788764000199121497980, 10.04029695948633799637872329337, 11.35670717936622720270022608472, 12.16112850918461978176616659966
