L(s) = 1 | + (−1 − 1.73i)2-s + (−1.99 + 3.46i)4-s + (−5.22 − 9.04i)5-s + 7.99·8-s + (−10.4 + 18.0i)10-s + (−30.5 + 52.9i)11-s + 59.2·13-s + (−8 − 13.8i)16-s + (10.2 − 17.7i)17-s + (−40.1 − 69.5i)19-s + 41.7·20-s + 122.·22-s + (79.1 + 137. i)23-s + (7.93 − 13.7i)25-s + (−59.2 − 102. i)26-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.467 − 0.809i)5-s + 0.353·8-s + (−0.330 + 0.572i)10-s + (−0.837 + 1.45i)11-s + 1.26·13-s + (−0.125 − 0.216i)16-s + (0.145 − 0.252i)17-s + (−0.485 − 0.840i)19-s + 0.467·20-s + 1.18·22-s + (0.717 + 1.24i)23-s + (0.0635 − 0.109i)25-s + (−0.446 − 0.773i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.6863404268\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6863404268\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + (5.22 + 9.04i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (30.5 - 52.9i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 - 59.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + (-10.2 + 17.7i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (40.1 + 69.5i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-79.1 - 137. i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + 85.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-121. + 210. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (145. + 251. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 - 168T + 6.89e4T^{2} \) |
| 43 | \( 1 - 7.62T + 7.95e4T^{2} \) |
| 47 | \( 1 + (84.6 + 146. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-125. + 216. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (402. - 697. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-16.5 - 28.7i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-138. + 240. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 631.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (384. - 665. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-209. - 362. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 - 761.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (786. + 1.36e3i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.04e3T + 9.12e5T^{2} \) |
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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
−9.309273737731035592443770095164, −8.673134572113198719152020687822, −7.76906692296531350624588059813, −7.08383712971509045926098722312, −5.64555839740390114981233787666, −4.66867651908309278526935639981, −3.92997883640711184495508414464, −2.61291499748378412180360482372, −1.44464485161264357297756313280, −0.23191399094423486143631332551,
1.14831987697699292890258770638, 2.91672180158096617827312958212, 3.70806217806283388848049930181, 5.05820013297250289599540558898, 6.13009673672508674707719343371, 6.58913927395881682346007454728, 7.78112821738884533244455838025, 8.354240758683671360369062642362, 9.001492270502736566574992466945, 10.56074820392561773769429251380