L(s) = 1 | + (1.58 + 2.73i)2-s + (−3.54 − 6.13i)3-s + (−0.997 + 1.72i)4-s − 13.6·5-s + (11.2 − 19.4i)6-s + (−7.16 + 12.4i)7-s + 18.9·8-s + (−11.6 + 20.1i)9-s + (−21.5 − 37.2i)10-s + (33.8 + 58.6i)11-s + 14.1·12-s − 45.3·14-s + (48.2 + 83.5i)15-s + (37.9 + 65.7i)16-s + (−0.168 + 0.292i)17-s − 73.5·18-s + ⋯ |
L(s) = 1 | + (0.558 + 0.967i)2-s + (−0.682 − 1.18i)3-s + (−0.124 + 0.215i)4-s − 1.21·5-s + (0.762 − 1.32i)6-s + (−0.386 + 0.670i)7-s + 0.839·8-s + (−0.430 + 0.745i)9-s + (−0.679 − 1.17i)10-s + (0.928 + 1.60i)11-s + 0.340·12-s − 0.864·14-s + (0.829 + 1.43i)15-s + (0.593 + 1.02i)16-s + (−0.00240 + 0.00417i)17-s − 0.962·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.309 - 0.950i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.309 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.692687 + 0.953946i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.692687 + 0.953946i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 \) |
good | 2 | \( 1 + (-1.58 - 2.73i)T + (-4 + 6.92i)T^{2} \) |
| 3 | \( 1 + (3.54 + 6.13i)T + (-13.5 + 23.3i)T^{2} \) |
| 5 | \( 1 + 13.6T + 125T^{2} \) |
| 7 | \( 1 + (7.16 - 12.4i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (-33.8 - 58.6i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 17 | \( 1 + (0.168 - 0.292i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (20.2 - 35.0i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-77.8 - 134. i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-16.8 - 29.2i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + 157.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (-29.3 - 50.7i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (29.6 + 51.3i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-104. + 180. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + 221.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 409.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (86.7 - 150. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (280. - 485. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-134. - 233. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + (30.4 - 52.8i)T + (-1.78e5 - 3.09e5i)T^{2} \) |
| 73 | \( 1 - 282.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 984.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.20e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-269. - 467. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (793. - 1.37e3i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−12.44423933890769324772949803769, −12.09183278439736254376896418203, −11.03710570690334846714235912080, −9.407776691346512177441807137943, −7.81964894010515806698840080902, −7.16974998828886022943151388180, −6.46755597882488392520340642372, −5.31959947716326335784430926109, −4.00970907460292610447460464367, −1.59203785230204965261393429457,
0.52243126721513021669868818410, 3.33709390935274612056950889583, 3.93180304815881742172388046337, 4.85109010167840236352875507493, 6.50888087201839680824291913402, 7.961237326732023995715243045234, 9.292290548761044842283449637189, 10.66389851580807739704785657952, 11.05113255367863619053347690216, 11.68165925969317224815209892307