| L(s) = 1 | − 16.5·2-s + 27·3-s + 144.·4-s − 425.·5-s − 445.·6-s − 1.50e3·7-s − 270.·8-s + 729·9-s + 7.02e3·10-s − 566.·11-s + 3.89e3·12-s − 1.04e4·13-s + 2.47e4·14-s − 1.14e4·15-s − 1.40e4·16-s + 8.18e3·17-s − 1.20e4·18-s − 4.87e4·19-s − 6.14e4·20-s − 4.05e4·21-s + 9.35e3·22-s + 1.21e4·23-s − 7.29e3·24-s + 1.02e5·25-s + 1.73e5·26-s + 1.96e4·27-s − 2.16e5·28-s + ⋯ |
| L(s) = 1 | − 1.45·2-s + 0.577·3-s + 1.12·4-s − 1.52·5-s − 0.842·6-s − 1.65·7-s − 0.186·8-s + 0.333·9-s + 2.22·10-s − 0.128·11-s + 0.651·12-s − 1.32·13-s + 2.41·14-s − 0.878·15-s − 0.855·16-s + 0.404·17-s − 0.486·18-s − 1.63·19-s − 1.71·20-s − 0.955·21-s + 0.187·22-s + 0.208·23-s − 0.107·24-s + 1.31·25-s + 1.93·26-s + 0.192·27-s − 1.86·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.2407390265\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2407390265\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 27T \) |
| 23 | \( 1 - 1.21e4T \) |
| good | 2 | \( 1 + 16.5T + 128T^{2} \) |
| 5 | \( 1 + 425.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.50e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 566.T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.04e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 8.18e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.87e4T + 8.93e8T^{2} \) |
| 29 | \( 1 + 7.85e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.82e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.97e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 5.67e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 7.51e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 7.48e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 3.77e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.10e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 7.92e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.41e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.72e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.33e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 7.94e5T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.22e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 5.72e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 7.01e6T + 8.07e13T^{2} \) |
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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
−13.00315893150941151247969652315, −12.08121984060091362637538856187, −10.61108168293431568342193221288, −9.688179708307984094221006976024, −8.680603763942186525545959954656, −7.64652025630109416054906330077, −6.81499237963137738237818862491, −4.12392635385709258768362526194, −2.66847044944035791687862439662, −0.37616687599646011118164969407,
0.37616687599646011118164969407, 2.66847044944035791687862439662, 4.12392635385709258768362526194, 6.81499237963137738237818862491, 7.64652025630109416054906330077, 8.680603763942186525545959954656, 9.688179708307984094221006976024, 10.61108168293431568342193221288, 12.08121984060091362637538856187, 13.00315893150941151247969652315
