| L(s) = 1 | − 8.29·2-s + 9·3-s + 36.8·4-s + 32.2·5-s − 74.6·6-s − 139.·7-s − 40.4·8-s + 81·9-s − 267.·10-s − 301.·11-s + 331.·12-s − 965.·13-s + 1.15e3·14-s + 290.·15-s − 844.·16-s + 60.5·17-s − 672.·18-s + 1.94e3·19-s + 1.18e3·20-s − 1.25e3·21-s + 2.50e3·22-s + 484.·23-s − 363.·24-s − 2.08e3·25-s + 8.01e3·26-s + 729·27-s − 5.12e3·28-s + ⋯ |
| L(s) = 1 | − 1.46·2-s + 0.577·3-s + 1.15·4-s + 0.577·5-s − 0.847·6-s − 1.07·7-s − 0.223·8-s + 0.333·9-s − 0.846·10-s − 0.751·11-s + 0.665·12-s − 1.58·13-s + 1.57·14-s + 0.333·15-s − 0.824·16-s + 0.0508·17-s − 0.489·18-s + 1.23·19-s + 0.664·20-s − 0.619·21-s + 1.10·22-s + 0.190·23-s − 0.128·24-s − 0.667·25-s + 2.32·26-s + 0.192·27-s − 1.23·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.8266279556\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8266279556\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 103 | \( 1 + 1.06e4T \) |
| good | 2 | \( 1 + 8.29T + 32T^{2} \) |
| 5 | \( 1 - 32.2T + 3.12e3T^{2} \) |
| 7 | \( 1 + 139.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 301.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 965.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 60.5T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.94e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 484.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.92e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.13e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.71e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.59e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 6.91e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.56e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.80e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.44e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.83e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.34e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.66e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.97e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.04e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.27e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.06e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.41e5T + 8.58e9T^{2} \) |
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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
−10.23042318344127729225477352497, −9.667684493575118642447722861207, −9.375537026198262970700705849442, −7.979084782355382863745258680214, −7.43307602522182515004315672732, −6.32051595154452308450514447709, −4.87293368339510531775753748493, −3.02865867137971919900388745435, −2.12534869607693761378918157314, −0.59419021947318550645133784345,
0.59419021947318550645133784345, 2.12534869607693761378918157314, 3.02865867137971919900388745435, 4.87293368339510531775753748493, 6.32051595154452308450514447709, 7.43307602522182515004315672732, 7.979084782355382863745258680214, 9.375537026198262970700705849442, 9.667684493575118642447722861207, 10.23042318344127729225477352497
