| L(s) = 1 | + (−0.308 + 0.142i)2-s + (0.947 + 0.319i)3-s + (−1.22 + 1.43i)4-s + (1.54 + 0.929i)5-s + (−0.337 + 0.0367i)6-s + (0.0319 − 0.589i)7-s + (0.353 − 1.27i)8-s + (0.796 + 0.605i)9-s + (−0.609 − 0.0662i)10-s + (−0.750 + 4.57i)11-s + (−1.61 + 0.971i)12-s + (−0.956 + 0.727i)13-s + (0.0742 + 0.186i)14-s + (1.16 + 1.37i)15-s + (−0.537 − 3.27i)16-s + (0.157 + 2.90i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 + 0.100i)2-s + (0.547 + 0.184i)3-s + (−0.610 + 0.718i)4-s + (0.691 + 0.415i)5-s + (−0.137 + 0.0149i)6-s + (0.0120 − 0.222i)7-s + (0.124 − 0.449i)8-s + (0.265 + 0.201i)9-s + (−0.192 − 0.0209i)10-s + (−0.226 + 1.38i)11-s + (−0.466 + 0.280i)12-s + (−0.265 + 0.201i)13-s + (0.0198 + 0.0498i)14-s + (0.301 + 0.354i)15-s + (−0.134 − 0.819i)16-s + (0.0381 + 0.703i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.432 - 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.432 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.00017 + 0.629302i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.00017 + 0.629302i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.947 - 0.319i)T \) |
| 59 | \( 1 + (-2.44 + 7.28i)T \) |
| good | 2 | \( 1 + (0.308 - 0.142i)T + (1.29 - 1.52i)T^{2} \) |
| 5 | \( 1 + (-1.54 - 0.929i)T + (2.34 + 4.41i)T^{2} \) |
| 7 | \( 1 + (-0.0319 + 0.589i)T + (-6.95 - 0.756i)T^{2} \) |
| 11 | \( 1 + (0.750 - 4.57i)T + (-10.4 - 3.51i)T^{2} \) |
| 13 | \( 1 + (0.956 - 0.727i)T + (3.47 - 12.5i)T^{2} \) |
| 17 | \( 1 + (-0.157 - 2.90i)T + (-16.9 + 1.83i)T^{2} \) |
| 19 | \( 1 + (-5.21 + 4.94i)T + (1.02 - 18.9i)T^{2} \) |
| 23 | \( 1 + (-5.34 + 1.17i)T + (20.8 - 9.65i)T^{2} \) |
| 29 | \( 1 + (4.53 + 2.09i)T + (18.7 + 22.1i)T^{2} \) |
| 31 | \( 1 + (4.92 + 4.66i)T + (1.67 + 30.9i)T^{2} \) |
| 37 | \( 1 + (0.764 + 2.75i)T + (-31.7 + 19.0i)T^{2} \) |
| 41 | \( 1 + (2.79 + 0.614i)T + (37.2 + 17.2i)T^{2} \) |
| 43 | \( 1 + (0.804 + 4.90i)T + (-40.7 + 13.7i)T^{2} \) |
| 47 | \( 1 + (-5.30 + 3.18i)T + (22.0 - 41.5i)T^{2} \) |
| 53 | \( 1 + (0.142 - 0.0155i)T + (51.7 - 11.3i)T^{2} \) |
| 61 | \( 1 + (9.25 - 4.28i)T + (39.4 - 46.4i)T^{2} \) |
| 67 | \( 1 + (1.40 - 5.04i)T + (-57.4 - 34.5i)T^{2} \) |
| 71 | \( 1 + (0.0797 - 0.0479i)T + (33.2 - 62.7i)T^{2} \) |
| 73 | \( 1 + (2.00 + 5.03i)T + (-52.9 + 50.2i)T^{2} \) |
| 79 | \( 1 + (-7.76 + 2.61i)T + (62.8 - 47.8i)T^{2} \) |
| 83 | \( 1 + (8.21 - 12.1i)T + (-30.7 - 77.1i)T^{2} \) |
| 89 | \( 1 + (-11.9 - 5.53i)T + (57.6 + 67.8i)T^{2} \) |
| 97 | \( 1 + (6.18 - 15.5i)T + (-70.4 - 66.7i)T^{2} \) |
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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
−13.09434895159838562498398969765, −12.07385158299174827118213480514, −10.61071114000989805474261561712, −9.613457694932235821106605906424, −9.066313503064376954218437873228, −7.64836250820611116128689643355, −7.00053101236832032806481965813, −5.13721794069753494037548538142, −3.91735651872127571244121064983, −2.41594396553028655870298240102,
1.35492738265947295920668128844, 3.24353088352944635213284952259, 5.16907144527504689331475854314, 5.83198670437205533416515511118, 7.54790565924188700152841256003, 8.802061082421132931726567801797, 9.313887819453175128255891522997, 10.31207675905415811768601450950, 11.41587417349391352640465249727, 12.79336357636606188846254876468
