| L(s) = 1 | + 15.0·2-s + 98.4·4-s − 159.·5-s + 980.·7-s − 445.·8-s − 2.40e3·10-s + 3.39e3·11-s − 7.16e3·13-s + 1.47e4·14-s − 1.92e4·16-s + 1.22e4·17-s + 1.95e4·19-s − 1.57e4·20-s + 5.10e4·22-s − 1.03e5·23-s − 5.25e4·25-s − 1.07e5·26-s + 9.64e4·28-s + 1.69e5·29-s − 3.77e4·31-s − 2.33e5·32-s + 1.84e5·34-s − 1.56e5·35-s − 2.95e4·37-s + 2.94e5·38-s + 7.12e4·40-s − 3.75e5·41-s + ⋯ |
| L(s) = 1 | + 1.32·2-s + 0.768·4-s − 0.572·5-s + 1.08·7-s − 0.307·8-s − 0.760·10-s + 0.768·11-s − 0.905·13-s + 1.43·14-s − 1.17·16-s + 0.604·17-s + 0.654·19-s − 0.439·20-s + 1.02·22-s − 1.76·23-s − 0.672·25-s − 1.20·26-s + 0.830·28-s + 1.28·29-s − 0.227·31-s − 1.25·32-s + 0.803·34-s − 0.617·35-s − 0.0958·37-s + 0.870·38-s + 0.175·40-s − 0.849·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 59 | \( 1 + 2.05e5T \) |
| good | 2 | \( 1 - 15.0T + 128T^{2} \) |
| 5 | \( 1 + 159.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 980.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 3.39e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 7.16e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.22e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.95e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 1.03e5T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.69e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 3.77e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.95e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + 3.75e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 7.19e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 3.17e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 2.90e5T + 1.17e12T^{2} \) |
| 61 | \( 1 + 2.85e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.89e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.57e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.76e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.56e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 6.61e5T + 2.71e13T^{2} \) |
| 89 | \( 1 - 5.31e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.84e6T + 8.07e13T^{2} \) |
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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
−9.259410418901815360742634971440, −8.128981671374583335993996892588, −7.41405553230353855215190682182, −6.23464887914067040344351105284, −5.33688828048038911198497231339, −4.46842443504332065991258514018, −3.85090577020874408327404268640, −2.69331605719864040662205289897, −1.49184665751187722813609008011, 0,
1.49184665751187722813609008011, 2.69331605719864040662205289897, 3.85090577020874408327404268640, 4.46842443504332065991258514018, 5.33688828048038911198497231339, 6.23464887914067040344351105284, 7.41405553230353855215190682182, 8.128981671374583335993996892588, 9.259410418901815360742634971440
