L(s) = 1 | + 4.82·2-s − 29.9·3-s − 8.68·4-s − 25·5-s − 144.·6-s + 29.5·7-s − 196.·8-s + 654.·9-s − 120.·10-s − 121·11-s + 260.·12-s + 345.·13-s + 142.·14-s + 749.·15-s − 670.·16-s + 2.12e3·17-s + 3.16e3·18-s + 361·19-s + 217.·20-s − 886.·21-s − 584.·22-s − 282.·23-s + 5.88e3·24-s + 625·25-s + 1.66e3·26-s − 1.23e4·27-s − 257.·28-s + ⋯ |
L(s) = 1 | + 0.853·2-s − 1.92·3-s − 0.271·4-s − 0.447·5-s − 1.64·6-s + 0.228·7-s − 1.08·8-s + 2.69·9-s − 0.381·10-s − 0.301·11-s + 0.521·12-s + 0.567·13-s + 0.194·14-s + 0.859·15-s − 0.654·16-s + 1.78·17-s + 2.30·18-s + 0.229·19-s + 0.121·20-s − 0.438·21-s − 0.257·22-s − 0.111·23-s + 2.08·24-s + 0.200·25-s + 0.483·26-s − 3.25·27-s − 0.0619·28-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
\(L(3)\) |
\(\approx\) |
\(0.9047654313\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9047654313\) |
\(L(\frac{7}{2})\) |
|
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 | \( 1 - 4.82T + 32T^{2} \) |
| 3 | \( 1 + 29.9T + 243T^{2} \) |
| 7 | \( 1 - 29.5T + 1.68e4T^{2} \) |
| 13 | \( 1 - 345.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.12e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 282.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.15e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.15e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.95e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.72e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 5.46e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.80e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.19e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.16e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.24e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.93e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.11e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.35e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.56e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.44e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.95e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.57e5T + 8.58e9T^{2} \) |
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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
−9.487587543385505753538619119191, −8.138282446790885004450290591184, −7.24778088279792097866626758463, −6.27800339123389052802803533361, −5.47586526092051724192352789204, −5.17200357148613262952849014150, −4.13424729469429448130713915921, −3.41430321518355396143466980880, −1.44377982265690966157798419255, −0.43312702399212309942343900798,
0.43312702399212309942343900798, 1.44377982265690966157798419255, 3.41430321518355396143466980880, 4.13424729469429448130713915921, 5.17200357148613262952849014150, 5.47586526092051724192352789204, 6.27800339123389052802803533361, 7.24778088279792097866626758463, 8.138282446790885004450290591184, 9.487587543385505753538619119191