| L(s) = 1 | − 19.3·2-s + 27·3-s + 247.·4-s + 336.·5-s − 523.·6-s − 879.·7-s − 2.31e3·8-s + 729·9-s − 6.51e3·10-s + 1.33e3·11-s + 6.68e3·12-s + 460.·13-s + 1.70e4·14-s + 9.07e3·15-s + 1.32e4·16-s + 2.49e4·17-s − 1.41e4·18-s − 2.47e3·19-s + 8.31e4·20-s − 2.37e4·21-s − 2.57e4·22-s + 1.05e5·23-s − 6.25e4·24-s + 3.47e4·25-s − 8.92e3·26-s + 1.96e4·27-s − 2.17e5·28-s + ⋯ |
| L(s) = 1 | − 1.71·2-s + 0.577·3-s + 1.93·4-s + 1.20·5-s − 0.988·6-s − 0.968·7-s − 1.59·8-s + 0.333·9-s − 2.05·10-s + 0.301·11-s + 1.11·12-s + 0.0581·13-s + 1.65·14-s + 0.694·15-s + 0.806·16-s + 1.23·17-s − 0.570·18-s − 0.0826·19-s + 2.32·20-s − 0.559·21-s − 0.516·22-s + 1.80·23-s − 0.923·24-s + 0.445·25-s − 0.0996·26-s + 0.192·27-s − 1.87·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.115175005\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.115175005\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 27T \) |
| 11 | \( 1 - 1.33e3T \) |
| good | 2 | \( 1 + 19.3T + 128T^{2} \) |
| 5 | \( 1 - 336.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 879.T + 8.23e5T^{2} \) |
| 13 | \( 1 - 460.T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.49e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.47e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 1.05e5T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.28e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.10e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.94e3T + 9.49e10T^{2} \) |
| 41 | \( 1 - 5.03e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 9.61e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.33e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 2.03e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.71e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 5.48e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.71e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.22e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.67e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 3.99e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 5.79e4T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.03e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 7.10e6T + 8.07e13T^{2} \) |
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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
−15.56882316659074613889592382761, −14.04569847760486786185005745882, −12.67654380498515770665014639496, −10.74834237437520131266032869389, −9.608229722204437014445829817084, −9.153107922119312846568116185085, −7.51919985911852597013658284710, −6.16641085540008548894515054368, −2.75464643051211002311515262324, −1.14283042662437233346125214717,
1.14283042662437233346125214717, 2.75464643051211002311515262324, 6.16641085540008548894515054368, 7.51919985911852597013658284710, 9.153107922119312846568116185085, 9.608229722204437014445829817084, 10.74834237437520131266032869389, 12.67654380498515770665014639496, 14.04569847760486786185005745882, 15.56882316659074613889592382761
