| L(s) = 1 | − 10.4·2-s − 9·3-s + 76.6·4-s + 93.7·6-s + 101.·7-s − 465.·8-s + 81·9-s + 121·11-s − 689.·12-s − 463.·13-s − 1.05e3·14-s + 2.39e3·16-s − 1.57e3·17-s − 844.·18-s − 2.43e3·19-s − 911.·21-s − 1.26e3·22-s − 2.76e3·23-s + 4.18e3·24-s + 4.82e3·26-s − 729·27-s + 7.75e3·28-s − 5.69e3·29-s − 5.75e3·31-s − 1.00e4·32-s − 1.08e3·33-s + 1.63e4·34-s + ⋯ |
| L(s) = 1 | − 1.84·2-s − 0.577·3-s + 2.39·4-s + 1.06·6-s + 0.780·7-s − 2.56·8-s + 0.333·9-s + 0.301·11-s − 1.38·12-s − 0.760·13-s − 1.43·14-s + 2.33·16-s − 1.31·17-s − 0.614·18-s − 1.54·19-s − 0.450·21-s − 0.555·22-s − 1.09·23-s + 1.48·24-s + 1.40·26-s − 0.192·27-s + 1.86·28-s − 1.25·29-s − 1.07·31-s − 1.73·32-s − 0.174·33-s + 2.43·34-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
| \(L(3)\) |
\(\approx\) |
\(0.1929550690\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1929550690\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 10.4T + 32T^{2} \) |
| 7 | \( 1 - 101.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 463.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.57e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.43e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.76e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.69e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.75e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.75e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.92e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.08e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.31e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.28e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.15e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.99e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.09e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.62e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.98e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.60e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.18e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.06e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.18e5T + 8.58e9T^{2} \) |
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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
−9.512744660029267647473647561852, −8.575113845585681080957899533012, −8.042853318266504785630928475745, −6.99502959229109646737149036008, −6.49128498037699843792684233811, −5.28097499300145940891154803319, −4.03345279462945993360666020092, −2.18121050550581449311156144962, −1.74158580162678695201025243806, −0.26233103596358636065280223081,
0.26233103596358636065280223081, 1.74158580162678695201025243806, 2.18121050550581449311156144962, 4.03345279462945993360666020092, 5.28097499300145940891154803319, 6.49128498037699843792684233811, 6.99502959229109646737149036008, 8.042853318266504785630928475745, 8.575113845585681080957899533012, 9.512744660029267647473647561852
