| L(s) = 1 | + (−0.166 − 0.313i)2-s + (0.725 + 0.687i)3-s + (1.05 − 1.55i)4-s + (2.73 + 0.601i)5-s + (0.0948 − 0.341i)6-s + (−2.80 − 2.13i)7-s + (−1.36 − 0.148i)8-s + (0.0541 + 0.998i)9-s + (−0.265 − 0.956i)10-s + (−0.0355 + 0.0892i)11-s + (1.83 − 0.402i)12-s + (−0.304 + 5.62i)13-s + (−0.202 + 1.23i)14-s + (1.57 + 2.31i)15-s + (−1.20 − 3.02i)16-s + (3.74 − 2.84i)17-s + ⋯ |
| L(s) = 1 | + (−0.117 − 0.221i)2-s + (0.419 + 0.397i)3-s + (0.525 − 0.775i)4-s + (1.22 + 0.269i)5-s + (0.0387 − 0.139i)6-s + (−1.05 − 0.805i)7-s + (−0.483 − 0.0525i)8-s + (0.0180 + 0.332i)9-s + (−0.0840 − 0.302i)10-s + (−0.0107 + 0.0269i)11-s + (0.528 − 0.116i)12-s + (−0.0845 + 1.55i)13-s + (−0.0540 + 0.329i)14-s + (0.405 + 0.598i)15-s + (−0.301 − 0.757i)16-s + (0.908 − 0.690i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.404i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.914 + 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.41291 - 0.298233i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.41291 - 0.298233i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.725 - 0.687i)T \) |
| 59 | \( 1 + (7.48 + 1.71i)T \) |
| good | 2 | \( 1 + (0.166 + 0.313i)T + (-1.12 + 1.65i)T^{2} \) |
| 5 | \( 1 + (-2.73 - 0.601i)T + (4.53 + 2.09i)T^{2} \) |
| 7 | \( 1 + (2.80 + 2.13i)T + (1.87 + 6.74i)T^{2} \) |
| 11 | \( 1 + (0.0355 - 0.0892i)T + (-7.98 - 7.56i)T^{2} \) |
| 13 | \( 1 + (0.304 - 5.62i)T + (-12.9 - 1.40i)T^{2} \) |
| 17 | \( 1 + (-3.74 + 2.84i)T + (4.54 - 16.3i)T^{2} \) |
| 19 | \( 1 + (4.51 + 1.52i)T + (15.1 + 11.4i)T^{2} \) |
| 23 | \( 1 + (-4.89 - 2.94i)T + (10.7 + 20.3i)T^{2} \) |
| 29 | \( 1 + (3.10 - 5.85i)T + (-16.2 - 24.0i)T^{2} \) |
| 31 | \( 1 + (5.94 - 2.00i)T + (24.6 - 18.7i)T^{2} \) |
| 37 | \( 1 + (-6.02 + 0.655i)T + (36.1 - 7.95i)T^{2} \) |
| 41 | \( 1 + (8.43 - 5.07i)T + (19.2 - 36.2i)T^{2} \) |
| 43 | \( 1 + (1.69 + 4.24i)T + (-31.2 + 29.5i)T^{2} \) |
| 47 | \( 1 + (-0.206 + 0.0455i)T + (42.6 - 19.7i)T^{2} \) |
| 53 | \( 1 + (-1.00 + 3.61i)T + (-45.4 - 27.3i)T^{2} \) |
| 61 | \( 1 + (6.11 + 11.5i)T + (-34.2 + 50.4i)T^{2} \) |
| 67 | \( 1 + (-5.24 - 0.570i)T + (65.4 + 14.4i)T^{2} \) |
| 71 | \( 1 + (-10.3 + 2.27i)T + (64.4 - 29.8i)T^{2} \) |
| 73 | \( 1 + (1.28 - 7.81i)T + (-69.1 - 23.3i)T^{2} \) |
| 79 | \( 1 + (-8.03 + 7.61i)T + (4.27 - 78.8i)T^{2} \) |
| 83 | \( 1 + (-6.14 - 7.23i)T + (-13.4 + 81.9i)T^{2} \) |
| 89 | \( 1 + (2.59 - 4.90i)T + (-49.9 - 73.6i)T^{2} \) |
| 97 | \( 1 + (0.160 + 0.980i)T + (-91.9 + 30.9i)T^{2} \) |
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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
−12.80976700351115200932170369408, −11.29763446809482157263832439116, −10.46131168802627057085641209215, −9.574545940863355177151006008028, −9.256372864407338270890794645596, −7.06199458074485765664376054111, −6.45224743452533119799445100619, −5.11883498503993334406882337799, −3.32042075592310904626592268516, −1.87966617387031516198974445075,
2.28967027348763917856129165124, 3.32924357248278531751630290599, 5.73856191352215546673357834741, 6.30253593799072252575643348276, 7.69799252390861853816294476379, 8.657100938070338660247774432517, 9.580635290163934774525000573183, 10.61272849076386500274045417387, 12.23557061588842632805794291886, 12.87241956092788042102511278400
