| L(s) = 1 | + (−22.5 − 69.3i)2-s + (−883. − 642. i)3-s + (2.33e3 − 1.69e3i)4-s + (1.47e4 − 4.55e4i)5-s + (−2.46e4 + 7.57e4i)6-s + (3.24e5 − 2.35e5i)7-s + (−6.52e5 − 4.74e5i)8-s + (−1.23e5 − 3.81e5i)9-s − 3.48e6·10-s + (2.45e6 + 5.33e6i)11-s − 3.14e6·12-s + (3.32e5 + 1.02e6i)13-s + (−2.36e7 − 1.71e7i)14-s + (−4.23e7 + 3.07e7i)15-s + (−1.08e7 + 3.34e7i)16-s + (−2.97e7 + 9.15e7i)17-s + ⋯ |
| L(s) = 1 | + (−0.248 − 0.765i)2-s + (−0.699 − 0.508i)3-s + (0.284 − 0.206i)4-s + (0.423 − 1.30i)5-s + (−0.215 + 0.662i)6-s + (1.04 − 0.757i)7-s + (−0.880 − 0.639i)8-s + (−0.0776 − 0.239i)9-s − 1.10·10-s + (0.418 + 0.908i)11-s − 0.304·12-s + (0.0191 + 0.0588i)13-s + (−0.839 − 0.610i)14-s + (−0.958 + 0.696i)15-s + (−0.162 + 0.499i)16-s + (−0.299 + 0.920i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 - 0.240i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.970 - 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.175785 + 1.44199i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.175785 + 1.44199i\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-2.45e6 - 5.33e6i)T \) |
| good | 2 | \( 1 + (22.5 + 69.3i)T + (-6.62e3 + 4.81e3i)T^{2} \) |
| 3 | \( 1 + (883. + 642. i)T + (4.92e5 + 1.51e6i)T^{2} \) |
| 5 | \( 1 + (-1.47e4 + 4.55e4i)T + (-9.87e8 - 7.17e8i)T^{2} \) |
| 7 | \( 1 + (-3.24e5 + 2.35e5i)T + (2.99e10 - 9.21e10i)T^{2} \) |
| 13 | \( 1 + (-3.32e5 - 1.02e6i)T + (-2.45e14 + 1.78e14i)T^{2} \) |
| 17 | \( 1 + (2.97e7 - 9.15e7i)T + (-8.01e15 - 5.82e15i)T^{2} \) |
| 19 | \( 1 + (2.13e8 + 1.55e8i)T + (1.29e16 + 3.99e16i)T^{2} \) |
| 23 | \( 1 - 1.29e9T + 5.04e17T^{2} \) |
| 29 | \( 1 + (-3.99e9 + 2.90e9i)T + (3.17e18 - 9.75e18i)T^{2} \) |
| 31 | \( 1 + (-1.21e9 - 3.74e9i)T + (-1.97e19 + 1.43e19i)T^{2} \) |
| 37 | \( 1 + (7.43e9 - 5.40e9i)T + (7.52e19 - 2.31e20i)T^{2} \) |
| 41 | \( 1 + (4.99e9 + 3.63e9i)T + (2.85e20 + 8.79e20i)T^{2} \) |
| 43 | \( 1 + 2.32e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + (-4.19e10 - 3.04e10i)T + (1.68e21 + 5.19e21i)T^{2} \) |
| 53 | \( 1 + (-2.98e9 - 9.18e9i)T + (-2.10e22 + 1.53e22i)T^{2} \) |
| 59 | \( 1 + (-3.05e11 + 2.21e11i)T + (3.24e22 - 9.98e22i)T^{2} \) |
| 61 | \( 1 + (6.29e10 - 1.93e11i)T + (-1.30e23 - 9.51e22i)T^{2} \) |
| 67 | \( 1 + 4.38e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + (-6.37e11 + 1.96e12i)T + (-9.42e23 - 6.84e23i)T^{2} \) |
| 73 | \( 1 + (2.20e9 - 1.59e9i)T + (5.16e23 - 1.59e24i)T^{2} \) |
| 79 | \( 1 + (7.54e11 + 2.32e12i)T + (-3.77e24 + 2.74e24i)T^{2} \) |
| 83 | \( 1 + (-4.58e11 + 1.41e12i)T + (-7.17e24 - 5.21e24i)T^{2} \) |
| 89 | \( 1 - 3.13e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + (2.73e12 + 8.43e12i)T + (-5.44e25 + 3.95e25i)T^{2} \) |
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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
−17.16140324030969636109546926034, −15.04795606966984655298553687538, −12.98125617966395493824915259686, −11.94588974590887984082122946820, −10.66491734429341554232280200037, −8.926170235124470843235705396138, −6.63834101094521804022770282703, −4.74069624840147415004188952748, −1.63405959411113441454454886525, −0.799587254353982260520931684883,
2.61676933394590128697382100788, 5.42812658729629696755339923993, 6.72300000971788020201773665999, 8.517018733081181957433806470074, 10.79111097343358595626982784255, 11.58925821452705196474446759404, 14.28045757765972809894236051310, 15.27352284025962270547874681019, 16.65293639096413792363162934964, 17.69121952040741088786753419617
