| L(s) = 1 | + 13.4·2-s + 35.1·3-s + 52.0·4-s + 471.·6-s − 1.34e3·7-s − 1.01e3·8-s − 952.·9-s + 5.55e3·11-s + 1.83e3·12-s + 1.42e4·13-s − 1.80e4·14-s − 2.03e4·16-s − 8.93e3·17-s − 1.27e4·18-s + 4.01e4·19-s − 4.72e4·21-s + 7.45e4·22-s + 5.39e4·23-s − 3.57e4·24-s + 1.91e5·26-s − 1.10e5·27-s − 7.01e4·28-s − 1.68e5·29-s + 1.38e5·31-s − 1.42e5·32-s + 1.95e5·33-s − 1.19e5·34-s + ⋯ |
| L(s) = 1 | + 1.18·2-s + 0.751·3-s + 0.407·4-s + 0.891·6-s − 1.48·7-s − 0.703·8-s − 0.435·9-s + 1.25·11-s + 0.305·12-s + 1.79·13-s − 1.75·14-s − 1.24·16-s − 0.441·17-s − 0.516·18-s + 1.34·19-s − 1.11·21-s + 1.49·22-s + 0.924·23-s − 0.528·24-s + 2.13·26-s − 1.07·27-s − 0.603·28-s − 1.28·29-s + 0.837·31-s − 0.769·32-s + 0.945·33-s − 0.523·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| good | 2 | \( 1 - 13.4T + 128T^{2} \) |
| 3 | \( 1 - 35.1T + 2.18e3T^{2} \) |
| 7 | \( 1 + 1.34e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 5.55e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.42e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 8.93e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.01e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 5.39e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.68e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.38e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.82e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 1.38e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 1.46e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.04e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.67e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 3.69e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.37e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 5.25e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.97e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 5.82e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.95e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 2.81e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 9.39e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 4.02e6T + 8.07e13T^{2} \) |
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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
−9.214060166096292577866723084233, −8.372293266889889557436391257158, −6.86619235950433686016581024334, −6.26963298142545074297409052483, −5.48864401044225074427738223246, −4.09231813265233869155913611865, −3.32183517014273480043050301430, −3.07955995554884862135538989274, −1.42033635031524634492367382264, 0,
1.42033635031524634492367382264, 3.07955995554884862135538989274, 3.32183517014273480043050301430, 4.09231813265233869155913611865, 5.48864401044225074427738223246, 6.26963298142545074297409052483, 6.86619235950433686016581024334, 8.372293266889889557436391257158, 9.214060166096292577866723084233
