| L(s) = 1 | + 22.2·2-s − 27·3-s + 365.·4-s − 417.·5-s − 600.·6-s + 1.38e3·7-s + 5.28e3·8-s + 729·9-s − 9.28e3·10-s + 3.38e3·11-s − 9.87e3·12-s + 2.86e3·13-s + 3.08e4·14-s + 1.12e4·15-s + 7.06e4·16-s + 3.11e3·17-s + 1.62e4·18-s + 1.67e4·19-s − 1.52e5·20-s − 3.75e4·21-s + 7.53e4·22-s − 1.21e4·23-s − 1.42e5·24-s + 9.65e4·25-s + 6.36e4·26-s − 1.96e4·27-s + 5.08e5·28-s + ⋯ |
| L(s) = 1 | + 1.96·2-s − 0.577·3-s + 2.85·4-s − 1.49·5-s − 1.13·6-s + 1.53·7-s + 3.65·8-s + 0.333·9-s − 2.93·10-s + 0.767·11-s − 1.65·12-s + 0.361·13-s + 3.00·14-s + 0.863·15-s + 4.31·16-s + 0.153·17-s + 0.654·18-s + 0.558·19-s − 4.27·20-s − 0.883·21-s + 1.50·22-s − 0.208·23-s − 2.10·24-s + 1.23·25-s + 0.710·26-s − 0.192·27-s + 4.37·28-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
| \(L(4)\) |
\(\approx\) |
\(5.495069780\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.495069780\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 - 22.2T + 128T^{2} \) |
| 5 | \( 1 + 417.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.38e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 3.38e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 2.86e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 3.11e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.67e4T + 8.93e8T^{2} \) |
| 29 | \( 1 + 1.44e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.50e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 5.47e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 7.76e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 5.74e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 3.11e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.16e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.53e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.51e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 9.63e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.64e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 7.24e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.56e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 1.99e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 4.59e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 9.59e6T + 8.07e13T^{2} \) |
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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
−13.24041103220376624616237308174, −11.93386079296008100422166797668, −11.58728356293410905615171255717, −10.89806376901653094259792093813, −7.957785442650327236747276633831, −7.06977754456537901767730319298, −5.52977690444082765031690525623, −4.47241108871731158586797915148, −3.63861117363256976152373963898, −1.52507854561749048297830769008,
1.52507854561749048297830769008, 3.63861117363256976152373963898, 4.47241108871731158586797915148, 5.52977690444082765031690525623, 7.06977754456537901767730319298, 7.957785442650327236747276633831, 10.89806376901653094259792093813, 11.58728356293410905615171255717, 11.93386079296008100422166797668, 13.24041103220376624616237308174
