| L(s) = 1 | + 5.08·2-s − 25.2·3-s − 102.·4-s + 250.·5-s − 128.·6-s + 343·7-s − 1.16e3·8-s − 1.55e3·9-s + 1.27e3·10-s + 79.1·11-s + 2.57e3·12-s + 2.19e3·13-s + 1.74e3·14-s − 6.31e3·15-s + 7.13e3·16-s − 1.98e4·17-s − 7.87e3·18-s + 6.41e3·19-s − 2.55e4·20-s − 8.65e3·21-s + 402.·22-s + 1.08e5·23-s + 2.95e4·24-s − 1.54e4·25-s + 1.11e4·26-s + 9.43e4·27-s − 3.50e4·28-s + ⋯ |
| L(s) = 1 | + 0.449·2-s − 0.539·3-s − 0.798·4-s + 0.895·5-s − 0.242·6-s + 0.377·7-s − 0.807·8-s − 0.708·9-s + 0.402·10-s + 0.0179·11-s + 0.430·12-s + 0.277·13-s + 0.169·14-s − 0.483·15-s + 0.435·16-s − 0.980·17-s − 0.318·18-s + 0.214·19-s − 0.714·20-s − 0.203·21-s + 0.00805·22-s + 1.85·23-s + 0.435·24-s − 0.198·25-s + 0.124·26-s + 0.922·27-s − 0.301·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.785802204\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.785802204\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 343T \) |
| 13 | \( 1 - 2.19e3T \) |
| good | 2 | \( 1 - 5.08T + 128T^{2} \) |
| 3 | \( 1 + 25.2T + 2.18e3T^{2} \) |
| 5 | \( 1 - 250.T + 7.81e4T^{2} \) |
| 11 | \( 1 - 79.1T + 1.94e7T^{2} \) |
| 17 | \( 1 + 1.98e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 6.41e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 1.08e5T + 3.40e9T^{2} \) |
| 29 | \( 1 - 2.49e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 6.08e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.82e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 8.99e4T + 1.94e11T^{2} \) |
| 43 | \( 1 - 3.46e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 6.31e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 8.61e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.57e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.00e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 9.99e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.15e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.40e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 6.46e5T + 1.92e13T^{2} \) |
| 83 | \( 1 + 8.03e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 3.26e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 3.88e6T + 8.07e13T^{2} \) |
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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
−12.86038959640322929147036088790, −11.67720767433113062535744726771, −10.60837826624719479967937635795, −9.311522267387381990394745833106, −8.452175695944079779350966975176, −6.51725580744031961673914647043, −5.50329678897213519535866187153, −4.58280851150161278552839752666, −2.78861643359035278617693336246, −0.836010255614980589089843058380,
0.836010255614980589089843058380, 2.78861643359035278617693336246, 4.58280851150161278552839752666, 5.50329678897213519535866187153, 6.51725580744031961673914647043, 8.452175695944079779350966975176, 9.311522267387381990394745833106, 10.60837826624719479967937635795, 11.67720767433113062535744726771, 12.86038959640322929147036088790
