| L(s) = 1 | + (1.78 − 0.193i)2-s + (1.59 + 0.666i)3-s + (1.18 − 0.261i)4-s + (−1.49 − 1.97i)5-s + (2.98 + 0.877i)6-s + (0.141 + 0.355i)7-s + (−1.33 + 0.448i)8-s + (2.11 + 2.13i)9-s + (−3.05 − 3.22i)10-s + (−0.521 − 0.241i)11-s + (2.07 + 0.373i)12-s + (−1.30 + 0.213i)13-s + (0.321 + 0.606i)14-s + (−1.08 − 4.15i)15-s + (−4.49 + 2.07i)16-s + (0.232 + 0.0927i)17-s + ⋯ |
| L(s) = 1 | + (1.26 − 0.137i)2-s + (0.923 + 0.384i)3-s + (0.594 − 0.130i)4-s + (−0.670 − 0.881i)5-s + (1.21 + 0.358i)6-s + (0.0535 + 0.134i)7-s + (−0.470 + 0.158i)8-s + (0.704 + 0.710i)9-s + (−0.965 − 1.01i)10-s + (−0.157 − 0.0727i)11-s + (0.599 + 0.107i)12-s + (−0.360 + 0.0591i)13-s + (0.0860 + 0.162i)14-s + (−0.279 − 1.07i)15-s + (−1.12 + 0.519i)16-s + (0.0564 + 0.0224i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.28463 - 0.0757541i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.28463 - 0.0757541i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.59 - 0.666i)T \) |
| 59 | \( 1 + (7.67 + 0.308i)T \) |
| good | 2 | \( 1 + (-1.78 + 0.193i)T + (1.95 - 0.429i)T^{2} \) |
| 5 | \( 1 + (1.49 + 1.97i)T + (-1.33 + 4.81i)T^{2} \) |
| 7 | \( 1 + (-0.141 - 0.355i)T + (-5.08 + 4.81i)T^{2} \) |
| 11 | \( 1 + (0.521 + 0.241i)T + (7.12 + 8.38i)T^{2} \) |
| 13 | \( 1 + (1.30 - 0.213i)T + (12.3 - 4.15i)T^{2} \) |
| 17 | \( 1 + (-0.232 - 0.0927i)T + (12.3 + 11.6i)T^{2} \) |
| 19 | \( 1 + (2.59 - 3.82i)T + (-7.03 - 17.6i)T^{2} \) |
| 23 | \( 1 + (0.107 + 1.98i)T + (-22.8 + 2.48i)T^{2} \) |
| 29 | \( 1 + (-0.506 + 4.66i)T + (-28.3 - 6.23i)T^{2} \) |
| 31 | \( 1 + (-3.80 + 2.58i)T + (11.4 - 28.7i)T^{2} \) |
| 37 | \( 1 + (-3.87 + 11.5i)T + (-29.4 - 22.3i)T^{2} \) |
| 41 | \( 1 + (1.00 + 0.0547i)T + (40.7 + 4.43i)T^{2} \) |
| 43 | \( 1 + (-3.93 - 8.50i)T + (-27.8 + 32.7i)T^{2} \) |
| 47 | \( 1 + (-4.02 - 3.06i)T + (12.5 + 45.2i)T^{2} \) |
| 53 | \( 1 + (8.54 - 9.01i)T + (-2.86 - 52.9i)T^{2} \) |
| 61 | \( 1 + (-1.51 - 13.8i)T + (-59.5 + 13.1i)T^{2} \) |
| 67 | \( 1 + (-0.589 - 1.74i)T + (-53.3 + 40.5i)T^{2} \) |
| 71 | \( 1 + (-8.28 + 10.8i)T + (-18.9 - 68.4i)T^{2} \) |
| 73 | \( 1 + (-5.25 + 2.78i)T + (40.9 - 60.4i)T^{2} \) |
| 79 | \( 1 + (3.45 - 4.06i)T + (-12.7 - 77.9i)T^{2} \) |
| 83 | \( 1 + (-1.52 - 0.918i)T + (38.8 + 73.3i)T^{2} \) |
| 89 | \( 1 + (12.9 + 1.41i)T + (86.9 + 19.1i)T^{2} \) |
| 97 | \( 1 + (10.1 + 5.39i)T + (54.4 + 80.2i)T^{2} \) |
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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
−12.67422249521470480932789786883, −12.22518396379832024516992935331, −10.92858529314794634388366135251, −9.539511914592073043571595538512, −8.572695026889926683705582104610, −7.71404771061982244604639493229, −5.91619683250950381714585872870, −4.56902005615625889847668335628, −4.03704628020480930135016627564, −2.56510742140177011975604587451,
2.72239731153057540137666325918, 3.63378925264853305846767728690, 4.83529781742503059024898010504, 6.53426091123107085422375243863, 7.22684413489711131722694171322, 8.430696982677717173907051191058, 9.673093366505598053775481630908, 11.01463046236058988957703718442, 12.09029380679034755497183219295, 12.89633833670808751649444803702
