| L(s) = 1 | + (−31.9 − 1.56i)2-s − 214. i·3-s + (1.01e3 + 100. i)4-s + (2.62e3 + 1.69e3i)5-s + (−336. + 6.85e3i)6-s + (−2.11e4 − 2.11e4i)7-s + (−3.24e4 − 4.80e3i)8-s + 1.29e4·9-s + (−8.11e4 − 5.84e4i)10-s + (−4.61e4 + 4.61e4i)11-s + (2.15e4 − 2.18e5i)12-s + 4.80e5i·13-s + (6.43e5 + 7.09e5i)14-s + (3.64e5 − 5.62e5i)15-s + (1.02e6 + 2.04e5i)16-s + (1.33e6 − 1.33e6i)17-s + ⋯ |
| L(s) = 1 | + (−0.998 − 0.0490i)2-s − 0.883i·3-s + (0.995 + 0.0979i)4-s + (0.839 + 0.543i)5-s + (−0.0433 + 0.882i)6-s + (−1.25 − 1.25i)7-s + (−0.989 − 0.146i)8-s + 0.220·9-s + (−0.811 − 0.584i)10-s + (−0.286 + 0.286i)11-s + (0.0865 − 0.878i)12-s + 1.29i·13-s + (1.19 + 1.31i)14-s + (0.480 − 0.741i)15-s + (0.980 + 0.195i)16-s + (0.939 − 0.939i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.802 - 0.596i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.802 - 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.885291 + 0.292865i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.885291 + 0.292865i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (31.9 + 1.56i)T \) |
| 5 | \( 1 + (-2.62e3 - 1.69e3i)T \) |
| good | 3 | \( 1 + 214. iT - 5.90e4T^{2} \) |
| 7 | \( 1 + (2.11e4 + 2.11e4i)T + 2.82e8iT^{2} \) |
| 11 | \( 1 + (4.61e4 - 4.61e4i)T - 2.59e10iT^{2} \) |
| 13 | \( 1 - 4.80e5iT - 1.37e11T^{2} \) |
| 17 | \( 1 + (-1.33e6 + 1.33e6i)T - 2.01e12iT^{2} \) |
| 19 | \( 1 + (-4.29e5 + 4.29e5i)T - 6.13e12iT^{2} \) |
| 23 | \( 1 + (7.93e6 - 7.93e6i)T - 4.14e13iT^{2} \) |
| 29 | \( 1 + (1.91e7 - 1.91e7i)T - 4.20e14iT^{2} \) |
| 31 | \( 1 + 1.48e7T + 8.19e14T^{2} \) |
| 37 | \( 1 + 5.04e7iT - 4.80e15T^{2} \) |
| 41 | \( 1 - 1.12e8iT - 1.34e16T^{2} \) |
| 43 | \( 1 - 1.16e8T + 2.16e16T^{2} \) |
| 47 | \( 1 + (1.75e7 - 1.75e7i)T - 5.25e16iT^{2} \) |
| 53 | \( 1 + 1.93e8T + 1.74e17T^{2} \) |
| 59 | \( 1 + (1.52e8 + 1.52e8i)T + 5.11e17iT^{2} \) |
| 61 | \( 1 + (-6.27e8 - 6.27e8i)T + 7.13e17iT^{2} \) |
| 67 | \( 1 - 2.78e8T + 1.82e18T^{2} \) |
| 71 | \( 1 - 1.55e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 + (-1.08e9 + 1.08e9i)T - 4.29e18iT^{2} \) |
| 79 | \( 1 - 7.56e8iT - 9.46e18T^{2} \) |
| 83 | \( 1 - 5.62e9iT - 1.55e19T^{2} \) |
| 89 | \( 1 - 7.35e9T + 3.11e19T^{2} \) |
| 97 | \( 1 + (-1.10e9 + 1.10e9i)T - 7.37e19iT^{2} \) |
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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
−12.43801093431045027145506654892, −11.12436851605622805662670227519, −9.847465345381481735149670393852, −9.544074314520894646269836617600, −7.37536226813156863154807416299, −7.12625374596855695504772541821, −6.04348550083992188099894071324, −3.49711955821940897456946005021, −2.05992424263136433391883567925, −1.01814270345187014455796620710,
0.38559536197080791942930798047, 2.11601981062264705852732251966, 3.36852932185832846332633964362, 5.55094196849851038315267395883, 6.13349365287471249231091537508, 8.082814321036557633073615158695, 9.119926876313230495984752615624, 9.946755099994343247572442128212, 10.44397821223936718516278962307, 12.28671664134936011409562707674
