| L(s) = 1 | + 7.65·2-s + 9·3-s + 26.6·4-s + 68.9·6-s + 112.·7-s − 40.9·8-s + 81·9-s + 121·11-s + 239.·12-s + 849.·13-s + 859.·14-s − 1.16e3·16-s + 608.·17-s + 620.·18-s − 2.24e3·19-s + 1.00e3·21-s + 926.·22-s + 1.28e3·23-s − 368.·24-s + 6.50e3·26-s + 729·27-s + 2.99e3·28-s + 2.67e3·29-s + 3.83e3·31-s − 7.62e3·32-s + 1.08e3·33-s + 4.65e3·34-s + ⋯ |
| L(s) = 1 | + 1.35·2-s + 0.577·3-s + 0.832·4-s + 0.781·6-s + 0.865·7-s − 0.226·8-s + 0.333·9-s + 0.301·11-s + 0.480·12-s + 1.39·13-s + 1.17·14-s − 1.13·16-s + 0.510·17-s + 0.451·18-s − 1.42·19-s + 0.499·21-s + 0.408·22-s + 0.507·23-s − 0.130·24-s + 1.88·26-s + 0.192·27-s + 0.720·28-s + 0.591·29-s + 0.717·31-s − 1.31·32-s + 0.174·33-s + 0.691·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(7.257974862\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.257974862\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 121T \) |
| good | 2 | \( 1 - 7.65T + 32T^{2} \) |
| 7 | \( 1 - 112.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 849.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 608.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.24e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.28e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.67e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.83e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.20e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 618.T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.33e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 6.89e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.98e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.81e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.60e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.56e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.60e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.31e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.18e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.10e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.04e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.49e5T + 8.58e9T^{2} \) |
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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
−9.288786368302876884921880171717, −8.564303931753046040586301113106, −7.80886654686363773327851803824, −6.52297499500718517826271509885, −5.93948823510845780040564263589, −4.74251456605511734585258701776, −4.16318149070506979895158668237, −3.25177616744893809849334382101, −2.21801751132365068554016567357, −1.02918147679256438828060744589,
1.02918147679256438828060744589, 2.21801751132365068554016567357, 3.25177616744893809849334382101, 4.16318149070506979895158668237, 4.74251456605511734585258701776, 5.93948823510845780040564263589, 6.52297499500718517826271509885, 7.80886654686363773327851803824, 8.564303931753046040586301113106, 9.288786368302876884921880171717
