| L(s) = 1 | + (−4.38 − 0.476i)2-s + (1.94 − 2.28i)3-s + (11.1 + 2.45i)4-s + (−11.3 − 8.61i)5-s + (−9.60 + 9.09i)6-s + (−7.94 + 19.9i)7-s + (−14.3 − 4.83i)8-s + (−1.45 − 8.88i)9-s + (45.5 + 43.1i)10-s + (27.1 − 12.5i)11-s + (27.3 − 20.7i)12-s + (−9.16 + 55.9i)13-s + (44.3 − 83.6i)14-s + (−41.7 + 9.18i)15-s + (−22.4 − 10.3i)16-s + (6.79 + 17.0i)17-s + ⋯ |
| L(s) = 1 | + (−1.54 − 0.168i)2-s + (0.373 − 0.440i)3-s + (1.39 + 0.307i)4-s + (−1.01 − 0.770i)5-s + (−0.653 + 0.618i)6-s + (−0.429 + 1.07i)7-s + (−0.634 − 0.213i)8-s + (−0.0539 − 0.328i)9-s + (1.44 + 1.36i)10-s + (0.744 − 0.344i)11-s + (0.657 − 0.499i)12-s + (−0.195 + 1.19i)13-s + (0.846 − 1.59i)14-s + (−0.717 + 0.158i)15-s + (−0.350 − 0.161i)16-s + (0.0969 + 0.243i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.377i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.925 + 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.639485 - 0.125354i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.639485 - 0.125354i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.94 + 2.28i)T \) |
| 59 | \( 1 + (361. - 273. i)T \) |
| good | 2 | \( 1 + (4.38 + 0.476i)T + (7.81 + 1.71i)T^{2} \) |
| 5 | \( 1 + (11.3 + 8.61i)T + (33.4 + 120. i)T^{2} \) |
| 7 | \( 1 + (7.94 - 19.9i)T + (-249. - 235. i)T^{2} \) |
| 11 | \( 1 + (-27.1 + 12.5i)T + (861. - 1.01e3i)T^{2} \) |
| 13 | \( 1 + (9.16 - 55.9i)T + (-2.08e3 - 701. i)T^{2} \) |
| 17 | \( 1 + (-6.79 - 17.0i)T + (-3.56e3 + 3.37e3i)T^{2} \) |
| 19 | \( 1 + (-1.42 - 2.09i)T + (-2.53e3 + 6.37e3i)T^{2} \) |
| 23 | \( 1 + (-0.0699 + 1.28i)T + (-1.20e4 - 1.31e3i)T^{2} \) |
| 29 | \( 1 + (-83.2 + 9.04i)T + (2.38e4 - 5.24e3i)T^{2} \) |
| 31 | \( 1 + (-92.4 + 136. i)T + (-1.10e4 - 2.76e4i)T^{2} \) |
| 37 | \( 1 + (-415. + 139. i)T + (4.03e4 - 3.06e4i)T^{2} \) |
| 41 | \( 1 + (14.3 + 263. i)T + (-6.85e4 + 7.45e3i)T^{2} \) |
| 43 | \( 1 + (-366. - 169. i)T + (5.14e4 + 6.05e4i)T^{2} \) |
| 47 | \( 1 + (170. - 129. i)T + (2.77e4 - 1.00e5i)T^{2} \) |
| 53 | \( 1 + (-389. + 368. i)T + (8.06e3 - 1.48e5i)T^{2} \) |
| 61 | \( 1 + (-474. - 51.6i)T + (2.21e5 + 4.87e4i)T^{2} \) |
| 67 | \( 1 + (-38.2 - 12.8i)T + (2.39e5 + 1.82e5i)T^{2} \) |
| 71 | \( 1 + (371. - 282. i)T + (9.57e4 - 3.44e5i)T^{2} \) |
| 73 | \( 1 + (60.7 - 114. i)T + (-2.18e5 - 3.21e5i)T^{2} \) |
| 79 | \( 1 + (-665. - 783. i)T + (-7.97e4 + 4.86e5i)T^{2} \) |
| 83 | \( 1 + (114. - 69.0i)T + (2.67e5 - 5.05e5i)T^{2} \) |
| 89 | \( 1 + (-342. + 37.2i)T + (6.88e5 - 1.51e5i)T^{2} \) |
| 97 | \( 1 + (319. + 603. i)T + (-5.12e5 + 7.55e5i)T^{2} \) |
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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
−11.86810603003066794924394789097, −11.35005315495517661763110249984, −9.678706041194991121530669673520, −8.983643285063217401822261779494, −8.401814061126636396305123633059, −7.43190831428698520079628758723, −6.22393619266846520818066557364, −4.21113196272285886263065020357, −2.36521810209546771521017390794, −0.831128311749943944647308323150,
0.74077351478732642945832065139, 3.03927362073629457716305861409, 4.28426211583579745409249034373, 6.60473388633810178133180050956, 7.48240552889338549657768678076, 8.091142481159614821286925142086, 9.357825540632544880793795960332, 10.24171309275215676288428395756, 10.78177513191780066365730699543, 11.83730835537691206201038152262
