| L(s) = 1 | + 0.315·2-s + 82.9·3-s − 127.·4-s + 531.·5-s + 26.1·6-s − 349.·7-s − 80.7·8-s + 4.69e3·9-s + 167.·10-s + 3.33e3·11-s − 1.06e4·12-s − 8.58e3·13-s − 110.·14-s + 4.41e4·15-s + 1.63e4·16-s − 2.57e4·17-s + 1.48e3·18-s + 1.69e4·19-s − 6.79e4·20-s − 2.90e4·21-s + 1.05e3·22-s + 8.07e4·23-s − 6.70e3·24-s + 2.04e5·25-s − 2.70e3·26-s + 2.08e5·27-s + 4.47e4·28-s + ⋯ |
| L(s) = 1 | + 0.0279·2-s + 1.77·3-s − 0.999·4-s + 1.90·5-s + 0.0495·6-s − 0.385·7-s − 0.0557·8-s + 2.14·9-s + 0.0530·10-s + 0.754·11-s − 1.77·12-s − 1.08·13-s − 0.0107·14-s + 3.37·15-s + 0.997·16-s − 1.27·17-s + 0.0599·18-s + 0.568·19-s − 1.90·20-s − 0.683·21-s + 0.0210·22-s + 1.38·23-s − 0.0989·24-s + 2.61·25-s − 0.0302·26-s + 2.03·27-s + 0.385·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.437088976\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.437088976\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + 7.95e4T \) |
| good | 2 | \( 1 - 0.315T + 128T^{2} \) |
| 3 | \( 1 - 82.9T + 2.18e3T^{2} \) |
| 5 | \( 1 - 531.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 349.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 3.33e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 8.58e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.57e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.69e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 8.07e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 6.77e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.88e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.79e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 6.11e5T + 1.94e11T^{2} \) |
| 47 | \( 1 + 1.18e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 5.38e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.96e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.10e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 8.13e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.76e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.47e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 3.68e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 3.77e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 2.83e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 4.40e6T + 8.07e13T^{2} \) |
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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
−14.26580613645794764599792817278, −13.43652887765336357354855847692, −12.93036365644329662779733018634, −10.00176788422954423633010139137, −9.385316994943992918079487975766, −8.736537338840566664914420887778, −6.84665796552863875515136036201, −4.89401645888586937643917968379, −3.07878366081793291452126047674, −1.71186247415649031714820697881,
1.71186247415649031714820697881, 3.07878366081793291452126047674, 4.89401645888586937643917968379, 6.84665796552863875515136036201, 8.736537338840566664914420887778, 9.385316994943992918079487975766, 10.00176788422954423633010139137, 12.93036365644329662779733018634, 13.43652887765336357354855847692, 14.26580613645794764599792817278
