| L(s) = 1 | + (4.89 − 2.82i)2-s + (−23.6 − 13.0i)3-s + (15.9 − 27.7i)4-s + (−95.8 − 55.3i)5-s + (−152. + 3.16i)6-s + (−163. − 282. i)7-s − 181. i·8-s + (390. + 615. i)9-s − 626.·10-s + (541. − 312. i)11-s + (−739. + 447. i)12-s + (1.01e3 − 1.75e3i)13-s + (−1.59e3 − 923. i)14-s + (1.54e3 + 2.55e3i)15-s + (−512. − 886. i)16-s + 4.12e3i·17-s + ⋯ |
| L(s) = 1 | + (0.612 − 0.353i)2-s + (−0.876 − 0.481i)3-s + (0.249 − 0.433i)4-s + (−0.766 − 0.442i)5-s + (−0.706 + 0.0146i)6-s + (−0.475 − 0.823i)7-s − 0.353i·8-s + (0.535 + 0.844i)9-s − 0.626·10-s + (0.407 − 0.235i)11-s + (−0.427 + 0.258i)12-s + (0.461 − 0.799i)13-s + (−0.582 − 0.336i)14-s + (0.458 + 0.757i)15-s + (−0.125 − 0.216i)16-s + 0.839i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.738 + 0.673i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.738 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.421310 - 1.08697i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.421310 - 1.08697i\) |
| \(L(4)\) |
|
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 | \( 1 + (23.6 + 13.0i)T \) |
| 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
−16.68731846217994426617589859863, −15.77217395051018090421217787271, −13.81601456177163850825466937850, −12.65553618662750616231382745911, −11.61459622390617229864328116969, −10.29121891108836585047701171291, −7.69664809858457341670088372738, −5.95698820449372778167748373641, −4.02785149678213105018757182993, −0.75577935722495958755642441221,
3.73828622670048338695051379995, 5.61324880855561358044482459906, 7.14167281862408199451551996203, 9.478926970833758556453354135415, 11.50432953680973197061555607137, 12.08647730904914928012726851240, 14.06070429600773594131927022633, 15.74543957640731671612848904290, 15.93907958824787144746024713795, 17.69789604301638132233114662574
