| 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
−12.48939455383541278638440362957, −12.05724410965021807234442424163, −10.59314991330208356499668093387, −9.727656820252477694598270685343, −7.894676757310848279201367845298, −7.32757062603945281150360052475, −5.40105375362310649301283648481, −4.07077401261409878147493503539, −2.37695495792975730073581916095, −1.75438303560819369543451908239,
0.851209093889229299839449275781, 2.88925751063323864814365244925, 4.55281410076400652959993156448, 5.55977466132240161883249302464, 6.83329486807245190739288759682, 7.917653675093768460401873147762, 9.461735738660962726938133259556, 10.33254514745447278395720057008, 11.48449461558555245194001550551, 13.07041649634121664363782235248
