| L(s) = 1 | + (−72.0 − 99.2i)2-s + (361. + 1.11e3i)3-s + (−3.38e3 + 1.04e4i)4-s + (−1.30e3 − 947. i)5-s + (8.43e4 − 1.16e5i)6-s + (−1.14e5 − 3.72e4i)7-s + (7.98e5 − 2.59e5i)8-s + (−6.76e5 + 4.91e5i)9-s + 1.97e5i·10-s + (4.94e5 − 1.70e6i)11-s − 1.28e7·12-s + (−4.47e5 − 6.16e5i)13-s + (4.57e6 + 1.40e7i)14-s + (5.82e5 − 1.79e6i)15-s + (−4.70e7 − 3.41e7i)16-s + (1.82e7 − 2.51e7i)17-s + ⋯ |
| L(s) = 1 | + (−1.12 − 1.55i)2-s + (0.495 + 1.52i)3-s + (−0.825 + 2.54i)4-s + (−0.0834 − 0.0606i)5-s + (1.80 − 2.48i)6-s + (−0.975 − 0.316i)7-s + (3.04 − 0.990i)8-s + (−1.27 + 0.925i)9-s + 0.197i·10-s + (0.278 − 0.960i)11-s − 4.28·12-s + (−0.0927 − 0.127i)13-s + (0.607 + 1.86i)14-s + (0.0511 − 0.157i)15-s + (−2.80 − 2.03i)16-s + (0.756 − 1.04i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.915 + 0.403i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.915 + 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(0.0783755 - 0.372090i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0783755 - 0.372090i\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-4.94e5 + 1.70e6i)T \) |
| good | 2 | \( 1 + (72.0 + 99.2i)T + (-1.26e3 + 3.89e3i)T^{2} \) |
| 3 | \( 1 + (-361. - 1.11e3i)T + (-4.29e5 + 3.12e5i)T^{2} \) |
| 5 | \( 1 + (1.30e3 + 947. i)T + (7.54e7 + 2.32e8i)T^{2} \) |
| 7 | \( 1 + (1.14e5 + 3.72e4i)T + (1.11e10 + 8.13e9i)T^{2} \) |
| 13 | \( 1 + (4.47e5 + 6.16e5i)T + (-7.19e12 + 2.21e13i)T^{2} \) |
| 17 | \( 1 + (-1.82e7 + 2.51e7i)T + (-1.80e14 - 5.54e14i)T^{2} \) |
| 19 | \( 1 + (2.25e7 - 7.32e6i)T + (1.79e15 - 1.30e15i)T^{2} \) |
| 23 | \( 1 + 1.44e8T + 2.19e16T^{2} \) |
| 29 | \( 1 + (2.38e8 + 7.75e7i)T + (2.86e17 + 2.07e17i)T^{2} \) |
| 31 | \( 1 + (-9.11e8 + 6.62e8i)T + (2.43e17 - 7.49e17i)T^{2} \) |
| 37 | \( 1 + (4.32e8 - 1.33e9i)T + (-5.32e18 - 3.86e18i)T^{2} \) |
| 41 | \( 1 + (4.21e9 - 1.37e9i)T + (1.82e19 - 1.32e19i)T^{2} \) |
| 43 | \( 1 + 6.94e9iT - 3.99e19T^{2} \) |
| 47 | \( 1 + (8.38e8 + 2.58e9i)T + (-9.40e19 + 6.82e19i)T^{2} \) |
| 53 | \( 1 + (-7.57e8 + 5.50e8i)T + (1.51e20 - 4.67e20i)T^{2} \) |
| 59 | \( 1 + (3.29e9 - 1.01e10i)T + (-1.43e21 - 1.04e21i)T^{2} \) |
| 61 | \( 1 + (1.54e10 - 2.12e10i)T + (-8.20e20 - 2.52e21i)T^{2} \) |
| 67 | \( 1 - 1.10e10T + 8.18e21T^{2} \) |
| 71 | \( 1 + (-8.97e10 - 6.51e10i)T + (5.07e21 + 1.56e22i)T^{2} \) |
| 73 | \( 1 + (6.59e10 + 2.14e10i)T + (1.85e22 + 1.34e22i)T^{2} \) |
| 79 | \( 1 + (2.30e11 + 3.17e11i)T + (-1.82e22 + 5.61e22i)T^{2} \) |
| 83 | \( 1 + (-1.09e10 + 1.50e10i)T + (-3.30e22 - 1.01e23i)T^{2} \) |
| 89 | \( 1 + 9.26e11T + 2.46e23T^{2} \) |
| 97 | \( 1 + (7.21e11 - 5.23e11i)T + (2.14e23 - 6.59e23i)T^{2} \) |
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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
−16.86933276676438858166334913941, −16.05875421935128530724639380457, −13.70056354276767322463023679130, −11.81007708440210260958675448207, −10.32220574468245800475286294783, −9.645533565043862072676324001714, −8.374659171641521926416527570296, −3.97399472560413979930946396601, −2.93160354035995222604562331779, −0.24893771116509450903125027408,
1.57238341088695546620882387869, 6.14317310390638523361355348476, 7.13932929563437953481196229007, 8.319875753636493492825444993872, 9.757695601369396534261675878922, 12.67597128081344774234799672949, 14.18620290228626617274652216219, 15.39248500221612137712654278687, 16.99115278890430607825727926425, 18.04484829054512978421654191293
