| L(s) = 1 | + (−4.98 − 8.63i)2-s + (−121.5 − 70.1i)3-s + (462. − 800. i)4-s + (2.61e3 − 1.51e3i)5-s + 1.39e3i·6-s + (1.34e4 + 1.00e4i)7-s − 1.94e4·8-s + (9.84e3 + 1.70e4i)9-s + (−2.61e4 − 1.50e4i)10-s + (6.41e4 − 1.11e5i)11-s + (−1.12e5 + 6.48e4i)12-s − 4.12e5i·13-s + (1.99e4 − 1.66e5i)14-s − 4.24e5·15-s + (−3.76e5 − 6.51e5i)16-s + (9.96e3 + 5.75e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.155 − 0.269i)2-s + (−0.5 − 0.288i)3-s + (0.451 − 0.781i)4-s + (0.837 − 0.483i)5-s + 0.179i·6-s + (0.800 + 0.599i)7-s − 0.593·8-s + (0.166 + 0.288i)9-s + (−0.261 − 0.150i)10-s + (0.398 − 0.689i)11-s + (−0.451 + 0.260i)12-s − 1.11i·13-s + (0.0371 − 0.309i)14-s − 0.558·15-s + (−0.358 − 0.621i)16-s + (0.00701 + 0.00405i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.493 + 0.869i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.493 + 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.885480 - 1.52086i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.885480 - 1.52086i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (121.5 + 70.1i)T \) |
| 7 | \( 1 + (-1.34e4 - 1.00e4i)T \) |
| good | 2 | \( 1 + (4.98 + 8.63i)T + (-512 + 886. i)T^{2} \) |
| 5 | \( 1 + (-2.61e3 + 1.51e3i)T + (4.88e6 - 8.45e6i)T^{2} \) |
| 11 | \( 1 + (-6.41e4 + 1.11e5i)T + (-1.29e10 - 2.24e10i)T^{2} \) |
| 13 | \( 1 + 4.12e5iT - 1.37e11T^{2} \) |
| 17 | \( 1 + (-9.96e3 - 5.75e3i)T + (1.00e12 + 1.74e12i)T^{2} \) |
| 19 | \( 1 + (1.36e6 - 7.88e5i)T + (3.06e12 - 5.30e12i)T^{2} \) |
| 23 | \( 1 + (4.66e6 + 8.07e6i)T + (-2.07e13 + 3.58e13i)T^{2} \) |
| 29 | \( 1 + 2.68e6T + 4.20e14T^{2} \) |
| 31 | \( 1 + (-6.30e6 - 3.63e6i)T + (4.09e14 + 7.09e14i)T^{2} \) |
| 37 | \( 1 + (-5.42e7 - 9.40e7i)T + (-2.40e15 + 4.16e15i)T^{2} \) |
| 41 | \( 1 + 1.33e8iT - 1.34e16T^{2} \) |
| 43 | \( 1 + 1.04e6T + 2.16e16T^{2} \) |
| 47 | \( 1 + (-1.04e8 + 6.01e7i)T + (2.62e16 - 4.55e16i)T^{2} \) |
| 53 | \( 1 + (-2.92e8 + 5.06e8i)T + (-8.74e16 - 1.51e17i)T^{2} \) |
| 59 | \( 1 + (8.72e8 + 5.03e8i)T + (2.55e17 + 4.42e17i)T^{2} \) |
| 61 | \( 1 + (-6.53e8 + 3.77e8i)T + (3.56e17 - 6.17e17i)T^{2} \) |
| 67 | \( 1 + (1.06e9 - 1.84e9i)T + (-9.11e17 - 1.57e18i)T^{2} \) |
| 71 | \( 1 - 2.55e9T + 3.25e18T^{2} \) |
| 73 | \( 1 + (-2.13e9 - 1.23e9i)T + (2.14e18 + 3.72e18i)T^{2} \) |
| 79 | \( 1 + (-1.03e9 - 1.79e9i)T + (-4.73e18 + 8.19e18i)T^{2} \) |
| 83 | \( 1 - 5.24e9iT - 1.55e19T^{2} \) |
| 89 | \( 1 + (-3.07e9 + 1.77e9i)T + (1.55e19 - 2.70e19i)T^{2} \) |
| 97 | \( 1 + 1.54e10iT - 7.37e19T^{2} \) |
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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
−15.31275944960110235839094395748, −14.06527345815743614535672995222, −12.50828081244003116789340808734, −11.24547648660179800048202455209, −10.07080420542483183717377257348, −8.468300834117113771133776301423, −6.20623592516354652782147745505, −5.28460460598058767266545477514, −2.15442201723006423065917282800, −0.843167403110878071124009481481,
1.96628410638076226430817604075, 4.20707647340712334423037698473, 6.27475723251762389812021049148, 7.49965097339518588763610934692, 9.398139473861589785992660053121, 10.92426035686104830601956691565, 12.01706207795901713997551890319, 13.72540928596394164481463249372, 15.00698243685397941355098103000, 16.49037603812762227136291870189
