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
−19.68260641684975545949521366667, −18.06838290643544627156859625341, −17.27266573596214497114966786560, −15.54698362592867342019821661304, −14.39783293332180938813372277817, −12.24102350056800006734909006456, −10.52309899803940669198100436923, −9.603724711571852166404302421427, −6.22782184244619567810527710834, −4.65100890831111296238168884095,
0.44893712434558485067792076063, 4.95257317413535698095332210365, 6.99612113289914227177493513414, 9.136009842348606173820629460494, 11.55023061059670182913460947277, 12.70373749456173013578104169289, 13.72389828051310726590066504801, 16.57799583718817073035547604995, 17.23823464822616978225932657447, 18.27411829629614669807278247154