L(s) = 1 | − 0.609·2-s + 20.7·3-s − 31.6·4-s + 25·5-s − 12.6·6-s − 114.·7-s + 38.7·8-s + 186.·9-s − 15.2·10-s + 121·11-s − 655.·12-s − 557.·13-s + 69.5·14-s + 518.·15-s + 988.·16-s + 1.25e3·17-s − 113.·18-s + 361·19-s − 790.·20-s − 2.36e3·21-s − 73.7·22-s + 112.·23-s + 803.·24-s + 625·25-s + 339.·26-s − 1.17e3·27-s + 3.61e3·28-s + ⋯ |
L(s) = 1 | − 0.107·2-s + 1.32·3-s − 0.988·4-s + 0.447·5-s − 0.143·6-s − 0.880·7-s + 0.214·8-s + 0.767·9-s − 0.0481·10-s + 0.301·11-s − 1.31·12-s − 0.914·13-s + 0.0948·14-s + 0.594·15-s + 0.965·16-s + 1.05·17-s − 0.0826·18-s + 0.229·19-s − 0.442·20-s − 1.17·21-s − 0.0324·22-s + 0.0444·23-s + 0.284·24-s + 0.200·25-s + 0.0984·26-s − 0.309·27-s + 0.870·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.413281607\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.413281607\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 - 361T \) |
good | 2 | \( 1 + 0.609T + 32T^{2} \) |
| 3 | \( 1 - 20.7T + 243T^{2} \) |
| 7 | \( 1 + 114.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 557.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.25e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 112.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.05e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.08e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.13e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 4.33e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.25e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.37e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.55e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.15e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.19e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.16e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.33e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.07e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.08e5T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.55e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 7.20e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.17e5T + 8.58e9T^{2} \) |
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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
−9.217776031770829432562493327670, −8.511082416494424366519316489576, −7.77301840155980871335971027191, −6.85976807293717101959054230530, −5.68519051946769229799425915311, −4.76480255600973834211545046407, −3.59059989395426041306990929586, −3.08522489467572626350836938556, −1.92161264025351619017243374901, −0.63621819340938302046083251771,
0.63621819340938302046083251771, 1.92161264025351619017243374901, 3.08522489467572626350836938556, 3.59059989395426041306990929586, 4.76480255600973834211545046407, 5.68519051946769229799425915311, 6.85976807293717101959054230530, 7.77301840155980871335971027191, 8.511082416494424366519316489576, 9.217776031770829432562493327670