| L(s) = 1 | − 5.60·2-s + 14.2·3-s − 0.613·4-s − 25·5-s − 79.7·6-s − 144.·7-s + 182.·8-s − 40.5·9-s + 140.·10-s + 121·11-s − 8.72·12-s + 356.·13-s + 810.·14-s − 355.·15-s − 1.00e3·16-s − 1.63e3·17-s + 227.·18-s − 361·19-s + 15.3·20-s − 2.05e3·21-s − 677.·22-s − 1.08e3·23-s + 2.59e3·24-s + 625·25-s − 1.99e3·26-s − 4.03e3·27-s + 88.7·28-s + ⋯ |
| L(s) = 1 | − 0.990·2-s + 0.912·3-s − 0.0191·4-s − 0.447·5-s − 0.903·6-s − 1.11·7-s + 1.00·8-s − 0.166·9-s + 0.442·10-s + 0.301·11-s − 0.0174·12-s + 0.585·13-s + 1.10·14-s − 0.408·15-s − 0.980·16-s − 1.37·17-s + 0.165·18-s − 0.229·19-s + 0.00856·20-s − 1.01·21-s − 0.298·22-s − 0.426·23-s + 0.921·24-s + 0.200·25-s − 0.579·26-s − 1.06·27-s + 0.0213·28-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
| \(L(3)\) |
\(\approx\) |
\(0.5027268645\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5027268645\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 5.60T + 32T^{2} \) |
| 3 | \( 1 - 14.2T + 243T^{2} \) |
| 7 | \( 1 + 144.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 356.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.63e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 1.08e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 595.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.40e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 8.38e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 2.05e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.60e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.38e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.09e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.99e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.84e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.53e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.45e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.44e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.08e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.37e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.58e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.38e4T + 8.58e9T^{2} \) |
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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
−9.107183864732930262703117521814, −8.523036549551872276503635205916, −7.88517523049131898574033357587, −6.89918873951513948999613724365, −6.11635538852176919023172499881, −4.57786994059137558476592459111, −3.74353479124773045571197969640, −2.82727951113009474427864869981, −1.69082389001198052503283176720, −0.33572211689679202434786114203,
0.33572211689679202434786114203, 1.69082389001198052503283176720, 2.82727951113009474427864869981, 3.74353479124773045571197969640, 4.57786994059137558476592459111, 6.11635538852176919023172499881, 6.89918873951513948999613724365, 7.88517523049131898574033357587, 8.523036549551872276503635205916, 9.107183864732930262703117521814
