| L(s) = 1 | + (0.288 − 0.0313i)2-s + (0.974 − 1.43i)3-s + (−1.87 + 0.411i)4-s + (−0.965 − 1.27i)5-s + (0.235 − 0.442i)6-s + (−1.63 − 4.10i)7-s + (−1.07 + 0.362i)8-s + (−1.10 − 2.79i)9-s + (−0.317 − 0.335i)10-s + (5.62 + 2.60i)11-s + (−1.23 + 3.08i)12-s + (0.422 − 0.0692i)13-s + (−0.599 − 1.13i)14-s + (−2.75 + 0.144i)15-s + (3.17 − 1.47i)16-s + (3.28 + 1.31i)17-s + ⋯ |
| L(s) = 1 | + (0.203 − 0.0221i)2-s + (0.562 − 0.826i)3-s + (−0.935 + 0.205i)4-s + (−0.431 − 0.568i)5-s + (0.0962 − 0.180i)6-s + (−0.618 − 1.55i)7-s + (−0.380 + 0.128i)8-s + (−0.366 − 0.930i)9-s + (−0.100 − 0.106i)10-s + (1.69 + 0.785i)11-s + (−0.356 + 0.889i)12-s + (0.117 − 0.0192i)13-s + (−0.160 − 0.302i)14-s + (−0.712 + 0.0374i)15-s + (0.794 − 0.367i)16-s + (0.797 + 0.317i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.120 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.120 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.737746 - 0.832311i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.737746 - 0.832311i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.974 + 1.43i)T \) |
| 59 | \( 1 + (-7.63 + 0.832i)T \) |
| good | 2 | \( 1 + (-0.288 + 0.0313i)T + (1.95 - 0.429i)T^{2} \) |
| 5 | \( 1 + (0.965 + 1.27i)T + (-1.33 + 4.81i)T^{2} \) |
| 7 | \( 1 + (1.63 + 4.10i)T + (-5.08 + 4.81i)T^{2} \) |
| 11 | \( 1 + (-5.62 - 2.60i)T + (7.12 + 8.38i)T^{2} \) |
| 13 | \( 1 + (-0.422 + 0.0692i)T + (12.3 - 4.15i)T^{2} \) |
| 17 | \( 1 + (-3.28 - 1.31i)T + (12.3 + 11.6i)T^{2} \) |
| 19 | \( 1 + (3.16 - 4.67i)T + (-7.03 - 17.6i)T^{2} \) |
| 23 | \( 1 + (0.0588 + 1.08i)T + (-22.8 + 2.48i)T^{2} \) |
| 29 | \( 1 + (0.0717 - 0.659i)T + (-28.3 - 6.23i)T^{2} \) |
| 31 | \( 1 + (-0.299 + 0.202i)T + (11.4 - 28.7i)T^{2} \) |
| 37 | \( 1 + (-0.347 + 1.03i)T + (-29.4 - 22.3i)T^{2} \) |
| 41 | \( 1 + (-8.55 - 0.463i)T + (40.7 + 4.43i)T^{2} \) |
| 43 | \( 1 + (1.03 + 2.24i)T + (-27.8 + 32.7i)T^{2} \) |
| 47 | \( 1 + (1.27 + 0.968i)T + (12.5 + 45.2i)T^{2} \) |
| 53 | \( 1 + (1.07 - 1.13i)T + (-2.86 - 52.9i)T^{2} \) |
| 61 | \( 1 + (-1.12 - 10.3i)T + (-59.5 + 13.1i)T^{2} \) |
| 67 | \( 1 + (0.0547 + 0.162i)T + (-53.3 + 40.5i)T^{2} \) |
| 71 | \( 1 + (6.59 - 8.67i)T + (-18.9 - 68.4i)T^{2} \) |
| 73 | \( 1 + (-10.5 + 5.57i)T + (40.9 - 60.4i)T^{2} \) |
| 79 | \( 1 + (-3.64 + 4.29i)T + (-12.7 - 77.9i)T^{2} \) |
| 83 | \( 1 + (9.20 + 5.53i)T + (38.8 + 73.3i)T^{2} \) |
| 89 | \( 1 + (-15.3 - 1.66i)T + (86.9 + 19.1i)T^{2} \) |
| 97 | \( 1 + (4.48 + 2.38i)T + (54.4 + 80.2i)T^{2} \) |
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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
−12.56666403674263953515339753734, −11.98493445727647987816164331356, −10.18885569142946806717941150231, −9.279051385185554925136793070075, −8.294041514399763015454939468831, −7.34614251751396882893013905086, −6.26224118574699287223490357206, −4.19723721644410055599150802469, −3.73613316506088883624025471964, −1.04783591542542373389906099736,
2.99828044313551852939915207637, 3.92869327575210553121900529257, 5.33623590662008355786056778862, 6.41743077148170395982706824138, 8.282439653240545297252214585685, 9.184062847298555086519532456198, 9.492266334059421961964589720391, 11.01997223795452475694192159402, 11.93082405378592342238166869861, 13.08504837458573229644353946600
