| L(s) = 1 | + 15.0·2-s + 27·3-s + 97.9·4-s + 367.·5-s + 405.·6-s − 91.9·7-s − 451.·8-s + 729·9-s + 5.52e3·10-s + 1.33e3·11-s + 2.64e3·12-s + 5.11e3·13-s − 1.38e3·14-s + 9.91e3·15-s − 1.93e4·16-s − 8.83e3·17-s + 1.09e4·18-s + 1.33e4·19-s + 3.59e4·20-s − 2.48e3·21-s + 2.00e4·22-s − 7.07e4·23-s − 1.21e4·24-s + 5.67e4·25-s + 7.69e4·26-s + 1.96e4·27-s − 9.00e3·28-s + ⋯ |
| L(s) = 1 | + 1.32·2-s + 0.577·3-s + 0.765·4-s + 1.31·5-s + 0.767·6-s − 0.101·7-s − 0.311·8-s + 0.333·9-s + 1.74·10-s + 0.301·11-s + 0.441·12-s + 0.646·13-s − 0.134·14-s + 0.758·15-s − 1.17·16-s − 0.436·17-s + 0.442·18-s + 0.445·19-s + 1.00·20-s − 0.0584·21-s + 0.400·22-s − 1.21·23-s − 0.180·24-s + 0.726·25-s + 0.858·26-s + 0.192·27-s − 0.0775·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(4.469640980\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.469640980\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 27T \) |
| 11 | \( 1 - 1.33e3T \) |
| good | 2 | \( 1 - 15.0T + 128T^{2} \) |
| 5 | \( 1 - 367.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 91.9T + 8.23e5T^{2} \) |
| 13 | \( 1 - 5.11e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 8.83e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.33e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 7.07e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.23e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.16e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.45e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 5.86e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 4.70e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 8.18e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 2.08e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.96e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.31e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.43e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.63e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 5.76e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.00e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 8.42e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 3.80e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.53e7T + 8.07e13T^{2} \) |
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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
−14.73606842888056143887888784828, −13.71161178900797377999993644058, −13.32307773312174961098897398448, −11.88194900787675781031018796392, −10.06874079327326950994687170545, −8.807707209032929220632494108771, −6.57951661563364711818909767817, −5.40101921208311571535378750923, −3.69475491268798158751465551929, −2.06964773939259369133903744582,
2.06964773939259369133903744582, 3.69475491268798158751465551929, 5.40101921208311571535378750923, 6.57951661563364711818909767817, 8.807707209032929220632494108771, 10.06874079327326950994687170545, 11.88194900787675781031018796392, 13.32307773312174961098897398448, 13.71161178900797377999993644058, 14.73606842888056143887888784828
