| L(s) = 1 | + (−42.4 + 58.4i)2-s + (64.7 − 199. i)3-s + (−348. − 1.07e3i)4-s + (3.84e3 − 2.79e3i)5-s + (8.90e3 + 1.22e4i)6-s + (1.89e5 − 6.17e4i)7-s + (−2.04e5 − 6.63e4i)8-s + (3.94e5 + 2.86e5i)9-s + 3.43e5i·10-s + (−1.20e6 + 1.29e6i)11-s − 2.36e5·12-s + (−3.48e6 + 4.80e6i)13-s + (−4.46e6 + 1.37e7i)14-s + (−3.08e5 − 9.47e5i)15-s + (1.62e7 − 1.18e7i)16-s + (7.39e6 + 1.01e7i)17-s + ⋯ |
| L(s) = 1 | + (−0.663 + 0.913i)2-s + (0.0888 − 0.273i)3-s + (−0.0850 − 0.261i)4-s + (0.246 − 0.178i)5-s + (0.190 + 0.262i)6-s + (1.61 − 0.524i)7-s + (−0.778 − 0.252i)8-s + (0.742 + 0.539i)9-s + 0.343i·10-s + (−0.682 + 0.730i)11-s − 0.0791·12-s + (−0.722 + 0.994i)13-s + (−0.592 + 1.82i)14-s + (−0.0270 − 0.0832i)15-s + (0.970 − 0.705i)16-s + (0.306 + 0.421i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.121 - 0.992i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.121 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(0.971084 + 1.09728i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.971084 + 1.09728i\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (1.20e6 - 1.29e6i)T \) |
| good | 2 | \( 1 + (42.4 - 58.4i)T + (-1.26e3 - 3.89e3i)T^{2} \) |
| 3 | \( 1 + (-64.7 + 199. i)T + (-4.29e5 - 3.12e5i)T^{2} \) |
| 5 | \( 1 + (-3.84e3 + 2.79e3i)T + (7.54e7 - 2.32e8i)T^{2} \) |
| 7 | \( 1 + (-1.89e5 + 6.17e4i)T + (1.11e10 - 8.13e9i)T^{2} \) |
| 13 | \( 1 + (3.48e6 - 4.80e6i)T + (-7.19e12 - 2.21e13i)T^{2} \) |
| 17 | \( 1 + (-7.39e6 - 1.01e7i)T + (-1.80e14 + 5.54e14i)T^{2} \) |
| 19 | \( 1 + (-6.03e7 - 1.95e7i)T + (1.79e15 + 1.30e15i)T^{2} \) |
| 23 | \( 1 + 3.86e7T + 2.19e16T^{2} \) |
| 29 | \( 1 + (-5.82e8 + 1.89e8i)T + (2.86e17 - 2.07e17i)T^{2} \) |
| 31 | \( 1 + (-1.28e7 - 9.30e6i)T + (2.43e17 + 7.49e17i)T^{2} \) |
| 37 | \( 1 + (-1.45e9 - 4.47e9i)T + (-5.32e18 + 3.86e18i)T^{2} \) |
| 41 | \( 1 + (4.27e9 + 1.38e9i)T + (1.82e19 + 1.32e19i)T^{2} \) |
| 43 | \( 1 + 7.38e9iT - 3.99e19T^{2} \) |
| 47 | \( 1 + (5.22e8 - 1.60e9i)T + (-9.40e19 - 6.82e19i)T^{2} \) |
| 53 | \( 1 + (1.07e10 + 7.83e9i)T + (1.51e20 + 4.67e20i)T^{2} \) |
| 59 | \( 1 + (-9.46e9 - 2.91e10i)T + (-1.43e21 + 1.04e21i)T^{2} \) |
| 61 | \( 1 + (2.60e10 + 3.58e10i)T + (-8.20e20 + 2.52e21i)T^{2} \) |
| 67 | \( 1 + 3.23e10T + 8.18e21T^{2} \) |
| 71 | \( 1 + (-5.80e10 + 4.21e10i)T + (5.07e21 - 1.56e22i)T^{2} \) |
| 73 | \( 1 + (-1.74e11 + 5.67e10i)T + (1.85e22 - 1.34e22i)T^{2} \) |
| 79 | \( 1 + (-1.93e11 + 2.66e11i)T + (-1.82e22 - 5.61e22i)T^{2} \) |
| 83 | \( 1 + (3.54e11 + 4.87e11i)T + (-3.30e22 + 1.01e23i)T^{2} \) |
| 89 | \( 1 + 9.75e10T + 2.46e23T^{2} \) |
| 97 | \( 1 + (9.23e11 + 6.71e11i)T + (2.14e23 + 6.59e23i)T^{2} \) |
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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
−17.69171984855519675008829875334, −16.72785134088346556297375675153, −15.24834659696870476365103678738, −13.86220567541421617987632983645, −11.98106786874311850707182489012, −9.956999732930922098199175746626, −8.051471944776485638043938072372, −7.24222744380838594359696307411, −4.88776322818451927904297215288, −1.62010308813537452623737872452,
0.959645300671816360896708357185, 2.67044665387970437136499761460, 5.30038574453378276365355612638, 8.078656784716040179567006941366, 9.694188038381657591259042187405, 10.92616657247193033235250744839, 12.18241673474641933746176345636, 14.36507560306358426942785575752, 15.59655412584888373553536465670, 18.02240688513115123436345216764
