| L(s) = 1 | + (0.707 − 1.22i)2-s + (−0.999 − 1.73i)4-s + (−5.74 − 3.31i)5-s − 2.82·8-s + (−8.12 + 4.68i)10-s + (−2.37 − 4.11i)11-s + 15.2i·13-s + (−2.00 + 3.46i)16-s + (−3.25 + 1.88i)17-s + (−3.62 − 2.09i)19-s + 13.2i·20-s − 6.72·22-s + (−13.8 + 24.0i)23-s + (9.48 + 16.4i)25-s + (18.7 + 10.8i)26-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−1.14 − 0.663i)5-s − 0.353·8-s + (−0.812 + 0.468i)10-s + (−0.216 − 0.374i)11-s + 1.17i·13-s + (−0.125 + 0.216i)16-s + (−0.191 + 0.110i)17-s + (−0.190 − 0.110i)19-s + 0.663i·20-s − 0.305·22-s + (−0.602 + 1.04i)23-s + (0.379 + 0.657i)25-s + (0.720 + 0.415i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9217740783\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9217740783\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.707 + 1.22i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + (5.74 + 3.31i)T + (12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (2.37 + 4.11i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 - 15.2iT - 169T^{2} \) |
| 17 | \( 1 + (3.25 - 1.88i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (3.62 + 2.09i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (13.8 - 24.0i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + 3.51T + 841T^{2} \) |
| 31 | \( 1 + (-42.3 + 24.4i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-1.47 + 2.54i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 - 27.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 10.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-45.6 - 26.3i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-27.9 - 48.4i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-33.5 + 19.3i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-78.3 - 45.2i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-17.3 - 29.9i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 36.4T + 5.04e3T^{2} \) |
| 73 | \( 1 + (45.5 - 26.3i)T + (2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-16.8 + 29.2i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 - 127. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (43.5 + 25.1i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 101. iT - 9.40e3T^{2} \) |
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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
−10.06200055947061908032813175928, −9.192388141590384303687580515360, −8.408611559002747283723630192968, −7.62882417060806470436134448740, −6.50340051238039404987693781207, −5.41898667089045671501414065010, −4.31649505340004293399770739683, −3.91007734863863222852464762648, −2.52781077489288089435699294788, −1.09151748166526813216292605729,
0.31361254154237396522775822579, 2.59883205973950040642520141965, 3.57952353694507419417180154458, 4.46331688877763571588688955919, 5.47663603788625569465403184449, 6.57455033099203155548181961156, 7.26401682784675859745365888040, 8.075738702052207823594699162514, 8.603731422066333938645164384283, 10.04632834159786011010691470830
