| L(s) = 1 | + (0.291 − 0.0947i)2-s + (−33.5 − 24.3i)3-s + (−51.7 + 37.5i)4-s + (29.6 − 91.3i)5-s + (−12.0 − 3.92i)6-s + (−91.4 − 125. i)7-s + (−23.0 + 31.7i)8-s + (306. + 942. i)9-s − 29.4i·10-s + (941. − 940. i)11-s + 2.65e3·12-s + (−2.70e3 + 877. i)13-s + (−38.6 − 28.0i)14-s + (−3.22e3 + 2.34e3i)15-s + (1.26e3 − 3.87e3i)16-s + (−6.33e3 − 2.05e3i)17-s + ⋯ |
| L(s) = 1 | + (0.0364 − 0.0118i)2-s + (−1.24 − 0.902i)3-s + (−0.807 + 0.586i)4-s + (0.237 − 0.730i)5-s + (−0.0559 − 0.0181i)6-s + (−0.266 − 0.367i)7-s + (−0.0450 + 0.0619i)8-s + (0.420 + 1.29i)9-s − 0.0294i·10-s + (0.707 − 0.706i)11-s + 1.53·12-s + (−1.22 + 0.399i)13-s + (−0.0140 − 0.0102i)14-s + (−0.955 + 0.693i)15-s + (0.307 − 0.946i)16-s + (−1.29 − 0.419i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 + 0.472i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.881 + 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.113134 - 0.449973i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.113134 - 0.449973i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-941. + 940. i)T \) |
| good | 2 | \( 1 + (-0.291 + 0.0947i)T + (51.7 - 37.6i)T^{2} \) |
| 3 | \( 1 + (33.5 + 24.3i)T + (225. + 693. i)T^{2} \) |
| 5 | \( 1 + (-29.6 + 91.3i)T + (-1.26e4 - 9.18e3i)T^{2} \) |
| 7 | \( 1 + (91.4 + 125. i)T + (-3.63e4 + 1.11e5i)T^{2} \) |
| 13 | \( 1 + (2.70e3 - 877. i)T + (3.90e6 - 2.83e6i)T^{2} \) |
| 17 | \( 1 + (6.33e3 + 2.05e3i)T + (1.95e7 + 1.41e7i)T^{2} \) |
| 19 | \( 1 + (-4.99e3 + 6.87e3i)T + (-1.45e7 - 4.47e7i)T^{2} \) |
| 23 | \( 1 + 2.37e3T + 1.48e8T^{2} \) |
| 29 | \( 1 + (-2.40e4 - 3.31e4i)T + (-1.83e8 + 5.65e8i)T^{2} \) |
| 31 | \( 1 + (1.41e4 + 4.33e4i)T + (-7.18e8 + 5.21e8i)T^{2} \) |
| 37 | \( 1 + (115. - 84.2i)T + (7.92e8 - 2.44e9i)T^{2} \) |
| 41 | \( 1 + (-958. + 1.31e3i)T + (-1.46e9 - 4.51e9i)T^{2} \) |
| 43 | \( 1 + 5.36e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 + (3.51e4 + 2.55e4i)T + (3.33e9 + 1.02e10i)T^{2} \) |
| 53 | \( 1 + (2.78e4 + 8.58e4i)T + (-1.79e10 + 1.30e10i)T^{2} \) |
| 59 | \( 1 + (9.75e4 - 7.08e4i)T + (1.30e10 - 4.01e10i)T^{2} \) |
| 61 | \( 1 + (2.46e5 + 8.02e4i)T + (4.16e10 + 3.02e10i)T^{2} \) |
| 67 | \( 1 - 1.63e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + (-5.07e4 + 1.56e5i)T + (-1.03e11 - 7.52e10i)T^{2} \) |
| 73 | \( 1 + (-1.61e5 - 2.21e5i)T + (-4.67e10 + 1.43e11i)T^{2} \) |
| 79 | \( 1 + (3.09e5 - 1.00e5i)T + (1.96e11 - 1.42e11i)T^{2} \) |
| 83 | \( 1 + (-6.03e5 - 1.96e5i)T + (2.64e11 + 1.92e11i)T^{2} \) |
| 89 | \( 1 + 5.78e5T + 4.96e11T^{2} \) |
| 97 | \( 1 + (9.64e4 + 2.96e5i)T + (-6.73e11 + 4.89e11i)T^{2} \) |
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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
−18.27411829629614669807278247154, −17.23823464822616978225932657447, −16.57799583718817073035547604995, −13.72389828051310726590066504801, −12.70373749456173013578104169289, −11.55023061059670182913460947277, −9.136009842348606173820629460494, −6.99612113289914227177493513414, −4.95257317413535698095332210365, −0.44893712434558485067792076063,
4.65100890831111296238168884095, 6.22782184244619567810527710834, 9.603724711571852166404302421427, 10.52309899803940669198100436923, 12.24102350056800006734909006456, 14.39783293332180938813372277817, 15.54698362592867342019821661304, 17.27266573596214497114966786560, 18.06838290643544627156859625341, 19.68260641684975545949521366667
