| L(s) = 1 | + (−0.0767 − 1.41i)2-s + (−0.135 − 1.72i)3-s + (−0.0105 + 0.00114i)4-s + (−0.435 − 1.29i)5-s + (−2.43 + 0.324i)6-s + (−0.863 + 1.27i)7-s + (−0.456 − 2.78i)8-s + (−2.96 + 0.469i)9-s + (−1.79 + 0.716i)10-s + (4.10 + 0.903i)11-s + (0.00340 + 0.0180i)12-s + (1.27 + 1.08i)13-s + (1.86 + 1.12i)14-s + (−2.17 + 0.928i)15-s + (−3.92 + 0.864i)16-s + (−1.70 + 1.15i)17-s + ⋯ |
| L(s) = 1 | + (−0.0542 − 1.00i)2-s + (−0.0784 − 0.996i)3-s + (−0.00526 + 0.000572i)4-s + (−0.194 − 0.578i)5-s + (−0.993 + 0.132i)6-s + (−0.326 + 0.481i)7-s + (−0.161 − 0.984i)8-s + (−0.987 + 0.156i)9-s + (−0.568 + 0.226i)10-s + (1.23 + 0.272i)11-s + (0.000983 + 0.00520i)12-s + (0.354 + 0.300i)13-s + (0.499 + 0.300i)14-s + (−0.561 + 0.239i)15-s + (−0.981 + 0.216i)16-s + (−0.414 + 0.281i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.854 + 0.519i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.854 + 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.307077 - 1.09533i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.307077 - 1.09533i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.135 + 1.72i)T \) |
| 59 | \( 1 + (3.37 - 6.90i)T \) |
| good | 2 | \( 1 + (0.0767 + 1.41i)T + (-1.98 + 0.216i)T^{2} \) |
| 5 | \( 1 + (0.435 + 1.29i)T + (-3.98 + 3.02i)T^{2} \) |
| 7 | \( 1 + (0.863 - 1.27i)T + (-2.59 - 6.50i)T^{2} \) |
| 11 | \( 1 + (-4.10 - 0.903i)T + (9.98 + 4.61i)T^{2} \) |
| 13 | \( 1 + (-1.27 - 1.08i)T + (2.10 + 12.8i)T^{2} \) |
| 17 | \( 1 + (1.70 - 1.15i)T + (6.29 - 15.7i)T^{2} \) |
| 19 | \( 1 + (0.106 + 0.200i)T + (-10.6 + 15.7i)T^{2} \) |
| 23 | \( 1 + (2.98 + 2.82i)T + (1.24 + 22.9i)T^{2} \) |
| 29 | \( 1 + (-1.83 - 0.0992i)T + (28.8 + 3.13i)T^{2} \) |
| 31 | \( 1 + (-5.88 - 3.11i)T + (17.3 + 25.6i)T^{2} \) |
| 37 | \( 1 + (-9.18 - 1.50i)T + (35.0 + 11.8i)T^{2} \) |
| 41 | \( 1 + (0.553 + 0.584i)T + (-2.21 + 40.9i)T^{2} \) |
| 43 | \( 1 + (1.76 + 8.03i)T + (-39.0 + 18.0i)T^{2} \) |
| 47 | \( 1 + (4.56 + 1.53i)T + (37.4 + 28.4i)T^{2} \) |
| 53 | \( 1 + (-6.54 - 2.60i)T + (38.4 + 36.4i)T^{2} \) |
| 61 | \( 1 + (-2.05 + 0.111i)T + (60.6 - 6.59i)T^{2} \) |
| 67 | \( 1 + (15.0 - 2.47i)T + (63.4 - 21.3i)T^{2} \) |
| 71 | \( 1 + (2.35 - 6.99i)T + (-56.5 - 42.9i)T^{2} \) |
| 73 | \( 1 + (-4.63 + 7.70i)T + (-34.1 - 64.4i)T^{2} \) |
| 79 | \( 1 + (6.67 - 3.08i)T + (51.1 - 60.2i)T^{2} \) |
| 83 | \( 1 + (4.39 - 15.8i)T + (-71.1 - 42.7i)T^{2} \) |
| 89 | \( 1 + (0.138 - 2.56i)T + (-88.4 - 9.62i)T^{2} \) |
| 97 | \( 1 + (3.19 + 5.31i)T + (-45.4 + 85.7i)T^{2} \) |
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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
−12.06944522370981476849236625637, −11.75605571993829169041872564252, −10.53060079613518859494961548819, −9.237391000777603961801200002209, −8.431867425456208059612790551590, −6.86923577997371185252215657301, −6.15140438268618410476018737995, −4.21881891461556791450607278549, −2.61498613334060040119522856130, −1.22822984366681311024196791532,
3.13113003946179379472325688407, 4.42446428912158450571248442287, 5.95007760318436748860740401855, 6.66661868623459104205638243811, 7.901914787140841501780299732264, 8.992082039967100658643005804482, 10.03991878712767061942345764512, 11.19461013904744953738159909893, 11.68547347875432768044346296773, 13.50586328827668446249084472756
