| L(s) = 1 | + (−4 − 6.92i)2-s + (−31.9 + 55.4i)4-s + (−60 + 103. i)5-s + (−188.5 − 326. i)7-s + 511.·8-s + 960·10-s + (−300 − 519. i)11-s + (−2.68e3 + 4.64e3i)13-s + (−1.50e3 + 2.61e3i)14-s + (−2.04e3 − 3.54e3i)16-s + 1.21e4·17-s + 1.62e4·19-s + (−3.84e3 − 6.65e3i)20-s + (−2.40e3 + 4.15e3i)22-s + (−5.31e4 + 9.21e4i)23-s + ⋯ |
| L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.214 + 0.371i)5-s + (−0.207 − 0.359i)7-s + 0.353·8-s + 0.303·10-s + (−0.0679 − 0.117i)11-s + (−0.338 + 0.586i)13-s + (−0.146 + 0.254i)14-s + (−0.125 − 0.216i)16-s + 0.600·17-s + 0.542·19-s + (−0.107 − 0.185i)20-s + (−0.0480 + 0.0832i)22-s + (−0.911 + 1.57i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.7267780785\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7267780785\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (4 + 6.92i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (60 - 103. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 7 | \( 1 + (188.5 + 326. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (300 + 519. i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + (2.68e3 - 4.64e3i)T + (-3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 - 1.21e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.62e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + (5.31e4 - 9.21e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + (8.86e4 + 1.53e5i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 + (-1.34e5 + 2.32e5i)T + (-1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 - 1.14e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (-5.60e4 + 9.71e4i)T + (-9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (-5.75e4 - 9.96e4i)T + (-1.35e11 + 2.35e11i)T^{2} \) |
| 47 | \( 1 + (2.80e5 + 4.86e5i)T + (-2.53e11 + 4.38e11i)T^{2} \) |
| 53 | \( 1 + 1.78e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + (-8.93e5 + 1.54e6i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-6.53e5 - 1.13e6i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-1.00e6 + 1.74e6i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + 4.06e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.85e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + (5.18e5 + 8.98e5i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (4.60e6 + 7.97e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + 1.28e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (4.27e6 + 7.40e6i)T + (-4.03e13 + 6.99e13i)T^{2} \) |
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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
−11.34446944905441595028344397516, −10.00433250615023167958577517529, −9.474771934042722421893783139762, −7.992181887159115232837824351388, −7.22208764601542497163449714258, −5.75353525832497447920380624434, −4.18106449043079647355303136832, −3.14506292536126644782347375218, −1.72407931125173373286164402604, −0.24929120797511964794559288493,
1.08405529269815194028617865424, 2.84888616212590315655681007856, 4.52305346948679617215476760399, 5.58752546157234425827775567103, 6.73259419156521551575050356193, 7.912449557146421133918393660593, 8.700859770269613845902095674047, 9.815732773557353663084794744962, 10.68907168166362496288197155443, 12.16467726113880678666057524412
