| L(s) = 1 | + 10.4·2-s + 9·3-s + 78.0·4-s + 46.0·5-s + 94.4·6-s + 183.·7-s + 483.·8-s + 81·9-s + 483.·10-s − 536.·11-s + 702.·12-s − 601.·13-s + 1.92e3·14-s + 414.·15-s + 2.57e3·16-s − 1.94e3·17-s + 849.·18-s − 1.82e3·19-s + 3.59e3·20-s + 1.64e3·21-s − 5.62e3·22-s − 143.·23-s + 4.35e3·24-s − 1.00e3·25-s − 6.31e3·26-s + 729·27-s + 1.43e4·28-s + ⋯ |
| L(s) = 1 | + 1.85·2-s + 0.577·3-s + 2.44·4-s + 0.824·5-s + 1.07·6-s + 1.41·7-s + 2.67·8-s + 0.333·9-s + 1.52·10-s − 1.33·11-s + 1.40·12-s − 0.987·13-s + 2.62·14-s + 0.475·15-s + 2.51·16-s − 1.63·17-s + 0.618·18-s − 1.16·19-s + 2.01·20-s + 0.815·21-s − 2.47·22-s − 0.0566·23-s + 1.54·24-s − 0.320·25-s − 1.83·26-s + 0.192·27-s + 3.44·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(8.633002728\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.633002728\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 59 | \( 1 + 3.48e3T \) |
| good | 2 | \( 1 - 10.4T + 32T^{2} \) |
| 5 | \( 1 - 46.0T + 3.12e3T^{2} \) |
| 7 | \( 1 - 183.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 536.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 601.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.94e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.82e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 143.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.18e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.33e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.98e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.80e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.62e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.00e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.70e4T + 4.18e8T^{2} \) |
| 61 | \( 1 + 1.08e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.32e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 8.24e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.59e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.86e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.05e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.14e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.70e5T + 8.58e9T^{2} \) |
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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
−12.11777636748009842954853362535, −11.02737081007042308086036026653, −10.24328135833529243344313787390, −8.461776123085412903249320910277, −7.38683189863956555657663459835, −6.17694478639302551874156748635, −4.95315966675852746144470160854, −4.44173186132094496042242604062, −2.51246152752769520170686628833, −2.09147037149266073019168229759,
2.09147037149266073019168229759, 2.51246152752769520170686628833, 4.44173186132094496042242604062, 4.95315966675852746144470160854, 6.17694478639302551874156748635, 7.38683189863956555657663459835, 8.461776123085412903249320910277, 10.24328135833529243344313787390, 11.02737081007042308086036026653, 12.11777636748009842954853362535
