| L(s) = 1 | + (0.0877 + 0.0666i)2-s + (0.468 + 0.883i)3-s + (−0.531 − 1.91i)4-s + (0.442 + 0.419i)5-s + (−0.0178 + 0.108i)6-s + (2.33 − 2.74i)7-s + (0.162 − 0.408i)8-s + (−0.561 + 0.827i)9-s + (0.0108 + 0.0663i)10-s + (3.02 + 1.82i)11-s + (1.44 − 1.36i)12-s + (0.575 + 0.848i)13-s + (0.387 − 0.0852i)14-s + (−0.163 + 0.587i)15-s + (−3.36 + 2.02i)16-s + (−0.775 − 0.913i)17-s + ⋯ |
| L(s) = 1 | + (0.0620 + 0.0471i)2-s + (0.270 + 0.510i)3-s + (−0.265 − 0.957i)4-s + (0.198 + 0.187i)5-s + (−0.00727 + 0.0443i)6-s + (0.880 − 1.03i)7-s + (0.0574 − 0.144i)8-s + (−0.187 + 0.275i)9-s + (0.00343 + 0.0209i)10-s + (0.912 + 0.548i)11-s + (0.416 − 0.394i)12-s + (0.159 + 0.235i)13-s + (0.103 − 0.0227i)14-s + (−0.0421 + 0.151i)15-s + (−0.841 + 0.506i)16-s + (−0.188 − 0.221i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.219i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 + 0.219i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.34238 - 0.148928i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.34238 - 0.148928i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.468 - 0.883i)T \) |
| 59 | \( 1 + (4.40 + 6.29i)T \) |
| good | 2 | \( 1 + (-0.0877 - 0.0666i)T + (0.535 + 1.92i)T^{2} \) |
| 5 | \( 1 + (-0.442 - 0.419i)T + (0.270 + 4.99i)T^{2} \) |
| 7 | \( 1 + (-2.33 + 2.74i)T + (-1.13 - 6.90i)T^{2} \) |
| 11 | \( 1 + (-3.02 - 1.82i)T + (5.15 + 9.71i)T^{2} \) |
| 13 | \( 1 + (-0.575 - 0.848i)T + (-4.81 + 12.0i)T^{2} \) |
| 17 | \( 1 + (0.775 + 0.913i)T + (-2.75 + 16.7i)T^{2} \) |
| 19 | \( 1 + (3.75 - 1.73i)T + (12.3 - 14.4i)T^{2} \) |
| 23 | \( 1 + (-2.81 - 0.949i)T + (18.3 + 13.9i)T^{2} \) |
| 29 | \( 1 + (7.21 - 5.48i)T + (7.75 - 27.9i)T^{2} \) |
| 31 | \( 1 + (1.77 + 0.822i)T + (20.0 + 23.6i)T^{2} \) |
| 37 | \( 1 + (1.02 + 2.56i)T + (-26.8 + 25.4i)T^{2} \) |
| 41 | \( 1 + (-6.93 + 2.33i)T + (32.6 - 24.8i)T^{2} \) |
| 43 | \( 1 + (4.73 - 2.84i)T + (20.1 - 37.9i)T^{2} \) |
| 47 | \( 1 + (7.93 - 7.51i)T + (2.54 - 46.9i)T^{2} \) |
| 53 | \( 1 + (1.55 - 9.45i)T + (-50.2 - 16.9i)T^{2} \) |
| 61 | \( 1 + (-2.38 - 1.81i)T + (16.3 + 58.7i)T^{2} \) |
| 67 | \( 1 + (2.67 - 6.71i)T + (-48.6 - 46.0i)T^{2} \) |
| 71 | \( 1 + (1.08 - 1.02i)T + (3.84 - 70.8i)T^{2} \) |
| 73 | \( 1 + (-13.9 + 3.07i)T + (66.2 - 30.6i)T^{2} \) |
| 79 | \( 1 + (-0.722 + 1.36i)T + (-44.3 - 65.3i)T^{2} \) |
| 83 | \( 1 + (5.87 - 0.638i)T + (81.0 - 17.8i)T^{2} \) |
| 89 | \( 1 + (-13.4 + 10.2i)T + (23.8 - 85.7i)T^{2} \) |
| 97 | \( 1 + (-9.58 - 2.10i)T + (88.0 + 40.7i)T^{2} \) |
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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
−12.85227938073445373695334029524, −11.24415408515673170304232429715, −10.69991617605708282204440085469, −9.705053311506549057599613694876, −8.857281767156584593549387734813, −7.43196263253689411508724194119, −6.26820023437391539575479807238, −4.83385620728292665065915758344, −4.02345143079921925543739692136, −1.66526857057050948349339088857,
2.08777015411121591112690333227, 3.62712404380794889105007167260, 5.12188000278385774606880349773, 6.47840961327492509025440204158, 7.82371750834464900184492481807, 8.656091022032845331982106053055, 9.232628555085056851247061790698, 11.21753675388487934976843790183, 11.76107489997994898129334379259, 12.83168476381083271808772126637
