| L(s) = 1 | + (0.370 + 1.68i)2-s + (0.622 + 2.93i)3-s + (0.930 − 0.430i)4-s + (1.51 + 0.419i)5-s + (−4.71 + 2.13i)6-s + (−1.04 + 0.992i)7-s + (5.24 + 6.90i)8-s + (−8.22 + 3.65i)9-s + (−0.146 + 2.70i)10-s + (−8.37 + 7.11i)11-s + (1.84 + 2.46i)12-s + (21.3 − 7.18i)13-s + (−2.05 − 1.39i)14-s + (−0.291 + 4.69i)15-s + (−7.02 + 8.26i)16-s + (−5.08 + 5.37i)17-s + ⋯ |
| L(s) = 1 | + (0.185 + 0.842i)2-s + (0.207 + 0.978i)3-s + (0.232 − 0.107i)4-s + (0.302 + 0.0839i)5-s + (−0.785 + 0.355i)6-s + (−0.149 + 0.141i)7-s + (0.655 + 0.862i)8-s + (−0.914 + 0.405i)9-s + (−0.0146 + 0.270i)10-s + (−0.761 + 0.646i)11-s + (0.153 + 0.205i)12-s + (1.64 − 0.552i)13-s + (−0.147 − 0.0997i)14-s + (−0.0194 + 0.313i)15-s + (−0.438 + 0.516i)16-s + (−0.299 + 0.315i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.603 - 0.797i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.603 - 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.884395 + 1.77804i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.884395 + 1.77804i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.622 - 2.93i)T \) |
| 59 | \( 1 + (10.1 + 58.1i)T \) |
| good | 2 | \( 1 + (-0.370 - 1.68i)T + (-3.63 + 1.67i)T^{2} \) |
| 5 | \( 1 + (-1.51 - 0.419i)T + (21.4 + 12.8i)T^{2} \) |
| 7 | \( 1 + (1.04 - 0.992i)T + (2.65 - 48.9i)T^{2} \) |
| 11 | \( 1 + (8.37 - 7.11i)T + (19.5 - 119. i)T^{2} \) |
| 13 | \( 1 + (-21.3 + 7.18i)T + (134. - 102. i)T^{2} \) |
| 17 | \( 1 + (5.08 - 5.37i)T + (-15.6 - 288. i)T^{2} \) |
| 19 | \( 1 + (7.55 + 18.9i)T + (-262. + 248. i)T^{2} \) |
| 23 | \( 1 + (-3.14 - 28.9i)T + (-516. + 113. i)T^{2} \) |
| 29 | \( 1 + (-11.4 + 51.9i)T + (-763. - 353. i)T^{2} \) |
| 31 | \( 1 + (0.998 - 2.50i)T + (-697. - 660. i)T^{2} \) |
| 37 | \( 1 + (-31.4 - 23.9i)T + (366. + 1.31e3i)T^{2} \) |
| 41 | \( 1 + (0.720 - 6.62i)T + (-1.64e3 - 361. i)T^{2} \) |
| 43 | \( 1 + (-54.5 + 64.1i)T + (-299. - 1.82e3i)T^{2} \) |
| 47 | \( 1 + (8.76 - 2.43i)T + (1.89e3 - 1.13e3i)T^{2} \) |
| 53 | \( 1 + (-81.8 + 4.43i)T + (2.79e3 - 303. i)T^{2} \) |
| 61 | \( 1 + (-48.0 + 10.5i)T + (3.37e3 - 1.56e3i)T^{2} \) |
| 67 | \( 1 + (53.7 - 40.8i)T + (1.20e3 - 4.32e3i)T^{2} \) |
| 71 | \( 1 + (31.2 - 8.67i)T + (4.31e3 - 2.59e3i)T^{2} \) |
| 73 | \( 1 + (-26.8 + 39.5i)T + (-1.97e3 - 4.95e3i)T^{2} \) |
| 79 | \( 1 + (-5.62 - 34.3i)T + (-5.91e3 + 1.99e3i)T^{2} \) |
| 83 | \( 1 + (32.8 - 17.4i)T + (3.86e3 - 5.70e3i)T^{2} \) |
| 89 | \( 1 + (-21.7 + 98.7i)T + (-7.18e3 - 3.32e3i)T^{2} \) |
| 97 | \( 1 + (34.5 + 51.0i)T + (-3.48e3 + 8.74e3i)T^{2} \) |
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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
−13.29582037987218125288266810306, −11.51312400871609592415167002120, −10.74223664398897697616227067716, −9.866201249319029009865325583618, −8.618113258773236564850621038661, −7.70544242159599716314738860755, −6.22396172898772386740309168047, −5.49163667981968420916086683806, −4.18689123924894539124634583720, −2.47344225551953763700906416796,
1.23848112412044276333183364786, 2.58906463959026273875249372184, 3.82423322458225680747459663229, 5.87977574649355953234820495985, 6.80464191122097660285085774127, 8.028975425672313549175387318603, 9.001004333047065917906619462525, 10.55310355319496024228646177285, 11.16877268708890497994563482040, 12.20006133941437273974644337675
