| 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
−12.16467726113880678666057524412, −10.68907168166362496288197155443, −9.815732773557353663084794744962, −8.700859770269613845902095674047, −7.912449557146421133918393660593, −6.73259419156521551575050356193, −5.58752546157234425827775567103, −4.52305346948679617215476760399, −2.84888616212590315655681007856, −1.08405529269815194028617865424,
0.24929120797511964794559288493, 1.72407931125173373286164402604, 3.14506292536126644782347375218, 4.18106449043079647355303136832, 5.75353525832497447920380624434, 7.22208764601542497163449714258, 7.992181887159115232837824351388, 9.474771934042722421893783139762, 10.00433250615023167958577517529, 11.34446944905441595028344397516
