L(s) = 1 | + (4.98 + 8.63i)2-s + (13.2 + 22.9i)3-s + (14.2 − 24.7i)4-s + 86.0·5-s + (−132. + 228. i)6-s + (171.5 − 297. i)7-s + 1.56e3·8-s + (742. − 1.28e3i)9-s + (428. + 742. i)10-s + (−2.60e3 − 4.51e3i)11-s + 755.·12-s + (−7.31e3 − 3.03e3i)13-s + 3.42e3·14-s + (1.13e3 + 1.97e3i)15-s + (5.95e3 + 1.03e4i)16-s + (5.82e3 − 1.00e4i)17-s + ⋯ |
L(s) = 1 | + (0.440 + 0.763i)2-s + (0.283 + 0.490i)3-s + (0.111 − 0.192i)4-s + 0.307·5-s + (−0.249 + 0.432i)6-s + (0.188 − 0.327i)7-s + 1.07·8-s + (0.339 − 0.587i)9-s + (0.135 + 0.234i)10-s + (−0.590 − 1.02i)11-s + 0.126·12-s + (−0.923 − 0.383i)13-s + 0.333·14-s + (0.0871 + 0.150i)15-s + (0.363 + 0.630i)16-s + (0.287 − 0.497i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.124i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.992 + 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(3.15750 - 0.198110i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.15750 - 0.198110i\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-171.5 + 297. i)T \) |
| 13 | \( 1 + (7.31e3 + 3.03e3i)T \) |
good | 2 | \( 1 + (-4.98 - 8.63i)T + (-64 + 110. i)T^{2} \) |
| 3 | \( 1 + (-13.2 - 22.9i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 - 86.0T + 7.81e4T^{2} \) |
| 11 | \( 1 + (2.60e3 + 4.51e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 17 | \( 1 + (-5.82e3 + 1.00e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-2.26e4 + 3.91e4i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (1.12e4 + 1.95e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-3.47e4 - 6.02e4i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 + 5.87e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + (-1.67e5 - 2.90e5i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 + (-2.60e5 - 4.50e5i)T + (-9.73e10 + 1.68e11i)T^{2} \) |
| 43 | \( 1 + (2.32e5 - 4.02e5i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 - 9.22e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 2.20e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + (-1.78e5 + 3.08e5i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-2.64e5 + 4.58e5i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-1.34e6 - 2.32e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + (4.53e5 - 7.84e5i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 + 1.64e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.90e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 1.12e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (4.30e6 + 7.45e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + (-1.53e6 + 2.65e6i)T + (-4.03e13 - 6.99e13i)T^{2} \) |
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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
−13.07041649634121664363782235248, −11.48449461558555245194001550551, −10.33254514745447278395720057008, −9.461735738660962726938133259556, −7.917653675093768460401873147762, −6.83329486807245190739288759682, −5.55977466132240161883249302464, −4.55281410076400652959993156448, −2.88925751063323864814365244925, −0.851209093889229299839449275781,
1.75438303560819369543451908239, 2.37695495792975730073581916095, 4.07077401261409878147493503539, 5.40105375362310649301283648481, 7.32757062603945281150360052475, 7.894676757310848279201367845298, 9.727656820252477694598270685343, 10.59314991330208356499668093387, 12.05724410965021807234442424163, 12.48939455383541278638440362957