| L(s) = 1 | + (−1.86 + 1.56i)2-s + (−1.40 − 1.01i)3-s + (0.684 − 3.87i)4-s + (0.0603 − 0.165i)5-s + (4.21 − 0.303i)6-s + (0.340 + 0.588i)7-s + (2.36 + 4.09i)8-s + (0.939 + 2.84i)9-s + (0.146 + 0.403i)10-s + 5.07i·11-s + (−4.89 + 4.75i)12-s + (−0.848 − 2.33i)13-s + (−1.55 − 0.566i)14-s + (−0.252 + 0.171i)15-s + (−3.41 − 1.24i)16-s + (−1.07 + 2.95i)17-s + ⋯ |
| L(s) = 1 | + (−1.32 + 1.10i)2-s + (−0.810 − 0.586i)3-s + (0.342 − 1.93i)4-s + (0.0269 − 0.0741i)5-s + (1.71 − 0.123i)6-s + (0.128 + 0.222i)7-s + (0.835 + 1.44i)8-s + (0.313 + 0.949i)9-s + (0.0464 + 0.127i)10-s + 1.52i·11-s + (−1.41 + 1.37i)12-s + (−0.235 − 0.646i)13-s + (−0.416 − 0.151i)14-s + (−0.0652 + 0.0442i)15-s + (−0.854 − 0.311i)16-s + (−0.261 + 0.717i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.496 - 0.868i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.496 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.193923 + 0.334122i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.193923 + 0.334122i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.40 + 1.01i)T \) |
| 19 | \( 1 + (0.122 - 4.35i)T \) |
| good | 2 | \( 1 + (1.86 - 1.56i)T + (0.347 - 1.96i)T^{2} \) |
| 5 | \( 1 + (-0.0603 + 0.165i)T + (-3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.340 - 0.588i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 - 5.07iT - 11T^{2} \) |
| 13 | \( 1 + (0.848 + 2.33i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (1.07 - 2.95i)T + (-13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (0.413 + 0.0728i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-0.0433 + 0.245i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 - 9.69iT - 31T^{2} \) |
| 37 | \( 1 - 5.05iT - 37T^{2} \) |
| 41 | \( 1 + (-8.91 + 7.48i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.329 - 1.86i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-0.352 - 0.0621i)T + (44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (4.97 + 4.17i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (0.574 + 3.25i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (7.26 - 2.64i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (4.89 - 5.83i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (10.2 - 8.56i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (2.55 + 14.4i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-2.04 + 5.61i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (0.844 - 0.487i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (1.48 - 8.40i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-8.73 - 10.4i)T + (-16.8 + 95.5i)T^{2} \) |
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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
−12.85504131036455083538718259039, −12.15110566172979926152007742560, −10.63978454178989736008884830713, −10.08518936812536544329927555064, −8.811734052246018755783881052994, −7.74159969131299684694385246220, −7.04639421125818055735995652675, −6.01709945459651121279465996065, −4.95085069437632087397836249225, −1.61202404851202775768854012557,
0.63801981691009384345949693835, 2.85959993701283800515640726211, 4.37235876670099440361746848143, 6.09475660967961523334554216699, 7.51264850693446009845468795418, 8.899395922339049933060582647526, 9.453260563393060053447497589117, 10.64635962347737243580488985825, 11.19997961892020366922926702875, 11.78144839240413789584676646943
