| L(s) = 1 | + (−4.89 − 2.82i)2-s + (15.9 + 27.7i)4-s + (95.8 − 55.3i)5-s + (−163. + 282. i)7-s − 181. i·8-s − 626.·10-s + (−541. − 312. i)11-s + (1.01e3 + 1.75e3i)13-s + (1.59e3 − 923. i)14-s + (−512. + 886. i)16-s + 4.12e3i·17-s + 1.21e4·19-s + (3.06e3 + 1.77e3i)20-s + (1.76e3 + 3.06e3i)22-s + (1.85e4 − 1.07e4i)23-s + ⋯ |
| L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (0.766 − 0.442i)5-s + (−0.475 + 0.823i)7-s − 0.353i·8-s − 0.626·10-s + (−0.407 − 0.235i)11-s + (0.461 + 0.799i)13-s + (0.582 − 0.336i)14-s + (−0.125 + 0.216i)16-s + 0.839i·17-s + 1.77·19-s + (0.383 + 0.221i)20-s + (0.166 + 0.287i)22-s + (1.52 − 0.881i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 - 0.322i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.946 - 0.322i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.39438 + 0.231011i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.39438 + 0.231011i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (4.89 + 2.82i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-95.8 + 55.3i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 7 | \( 1 + (163. - 282. i)T + (-5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (541. + 312. i)T + (8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 + (-1.01e3 - 1.75e3i)T + (-2.41e6 + 4.18e6i)T^{2} \) |
| 17 | \( 1 - 4.12e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 1.21e4T + 4.70e7T^{2} \) |
| 23 | \( 1 + (-1.85e4 + 1.07e4i)T + (7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 + (-2.14e4 - 1.23e4i)T + (2.97e8 + 5.15e8i)T^{2} \) |
| 31 | \( 1 + (-2.02e4 - 3.51e4i)T + (-4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + 3.00e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + (5.12e4 - 2.96e4i)T + (2.37e9 - 4.11e9i)T^{2} \) |
| 43 | \( 1 + (-4.39e4 + 7.61e4i)T + (-3.16e9 - 5.47e9i)T^{2} \) |
| 47 | \( 1 + (1.12e5 + 6.48e4i)T + (5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 - 1.65e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + (4.61e4 - 2.66e4i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (-1.33e5 + 2.31e5i)T + (-2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-2.03e5 - 3.52e5i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 + 1.86e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 2.42e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + (-6.31e4 + 1.09e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + (-5.97e4 - 3.44e4i)T + (1.63e11 + 2.83e11i)T^{2} \) |
| 89 | \( 1 + 4.13e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (4.89e5 - 8.47e5i)T + (-4.16e11 - 7.21e11i)T^{2} \) |
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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
−13.96230492701391321895687769795, −12.88382941834367649677290472762, −11.86491170755740256925959226655, −10.49273144694762366173256450805, −9.316411375174365675345728105661, −8.552382194319143978721502987019, −6.71293380162610771328143079651, −5.28407073412654759847861225940, −3.00744885810876573145583567521, −1.34601394346812884647110319282,
0.870856977876909844757969776172, 2.98588262674021069536167170950, 5.33593178995569891443533345066, 6.73777859705335792550079352780, 7.79602024714311554289579078651, 9.543658502860764388203867186787, 10.18920761557483737743406435214, 11.43776675662403056035341702429, 13.27310846399026669710177564835, 13.94589460920070672807799394265
