| L(s) = 1 | + (1 − 1.73i)2-s + (−1.99 − 3.46i)4-s + (−2.29 + 3.97i)5-s − 7.99·8-s + (4.58 + 7.94i)10-s + (−3.24 − 5.61i)11-s − 45.2·13-s + (−8 + 13.8i)16-s + (−40.7 − 70.6i)17-s + (−2.52 + 4.37i)19-s + 18.3·20-s − 12.9·22-s + (53.1 − 92.0i)23-s + (51.9 + 90.0i)25-s + (−45.2 + 78.3i)26-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.205 + 0.355i)5-s − 0.353·8-s + (0.145 + 0.251i)10-s + (−0.0888 − 0.153i)11-s − 0.964·13-s + (−0.125 + 0.216i)16-s + (−0.581 − 1.00i)17-s + (−0.0305 + 0.0528i)19-s + 0.205·20-s − 0.125·22-s + (0.481 − 0.834i)23-s + (0.415 + 0.720i)25-s + (−0.341 + 0.590i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.661i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.394014384\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.394014384\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1 + 1.73i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + (2.29 - 3.97i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (3.24 + 5.61i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 45.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + (40.7 + 70.6i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (2.52 - 4.37i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-53.1 + 92.0i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 268.T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-146. - 253. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (57.2 - 99.2i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 161.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 471.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (173. - 299. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-202. - 351. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-126. - 219. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (375. - 650. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (5.82 + 10.0i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 681.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-342. - 593. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (0.132 - 0.228i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 437.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-29.2 + 50.6i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 1.28e3T + 9.12e5T^{2} \) |
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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
−10.12228599220177778800562557217, −9.091117290393997548983008528745, −8.315480264734920342970561557258, −7.08128515976452681167555377414, −6.52117307653332419221556341967, −5.06357129771516467069248785601, −4.63050279976944092648708099988, −3.17223837775400946580165782771, −2.57845861339206173677036291346, −1.04836784094789635516759232681,
0.36127376845765780076231228750, 2.08952014264613264108021015365, 3.38331683814579567936102602662, 4.52657336495654746081666719398, 5.09570395837229049823408273628, 6.30155404260583660587168673293, 6.95281073799891623302048060598, 8.055788958127365801970805526399, 8.506448691744356321990688754380, 9.627599008287067019858286065858
