| L(s) = 1 | + 20.0·2-s + 273.·4-s − 125·5-s − 343·7-s + 2.92e3·8-s − 2.50e3·10-s − 4.87e3·11-s + 1.11e4·13-s − 6.87e3·14-s + 2.35e4·16-s − 2.13e4·17-s − 5.02e4·19-s − 3.42e4·20-s − 9.77e4·22-s − 2.61e4·23-s + 1.56e4·25-s + 2.22e5·26-s − 9.38e4·28-s + 2.42e4·29-s + 3.00e4·31-s + 9.74e4·32-s − 4.28e5·34-s + 4.28e4·35-s − 4.80e5·37-s − 1.00e6·38-s − 3.65e5·40-s + 4.70e5·41-s + ⋯ |
| L(s) = 1 | + 1.77·2-s + 2.13·4-s − 0.447·5-s − 0.377·7-s + 2.01·8-s − 0.792·10-s − 1.10·11-s + 1.40·13-s − 0.669·14-s + 1.43·16-s − 1.05·17-s − 1.68·19-s − 0.956·20-s − 1.95·22-s − 0.448·23-s + 0.199·25-s + 2.48·26-s − 0.808·28-s + 0.184·29-s + 0.180·31-s + 0.525·32-s − 1.87·34-s + 0.169·35-s − 1.55·37-s − 2.97·38-s − 0.902·40-s + 1.06·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 125T \) |
| 7 | \( 1 + 343T \) |
| good | 2 | \( 1 - 20.0T + 128T^{2} \) |
| 11 | \( 1 + 4.87e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.11e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.13e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 5.02e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 2.61e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 2.42e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 3.00e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.80e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 4.70e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 4.12e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 2.75e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.52e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 3.06e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 8.07e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.08e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.54e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.89e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 4.26e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 6.70e5T + 2.71e13T^{2} \) |
| 89 | \( 1 - 2.44e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.38e7T + 8.07e13T^{2} \) |
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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
−10.75742802788491390456101660160, −8.873656289562443076827782673017, −7.81011094674362021497373189305, −6.56196096683021057290336478528, −6.01329628217670485361046907063, −4.76847956527270158066898794865, −3.99650039218718471553506630990, −3.01575189218956383468717371048, −1.94210246883825776132584819517, 0,
1.94210246883825776132584819517, 3.01575189218956383468717371048, 3.99650039218718471553506630990, 4.76847956527270158066898794865, 6.01329628217670485361046907063, 6.56196096683021057290336478528, 7.81011094674362021497373189305, 8.873656289562443076827782673017, 10.75742802788491390456101660160
