| L(s) = 1 | − 1.79·2-s − 0.0643·3-s + 1.23·4-s − 3.25·5-s + 0.115·6-s + 2.45·7-s + 1.36·8-s − 2.99·9-s + 5.86·10-s + 11-s − 0.0797·12-s − 3.20·13-s − 4.41·14-s + 0.209·15-s − 4.94·16-s + 3.74·17-s + 5.39·18-s − 6.81·19-s − 4.03·20-s − 0.157·21-s − 1.79·22-s + 7.26·23-s − 0.0880·24-s + 5.60·25-s + 5.76·26-s + 0.385·27-s + 3.04·28-s + ⋯ |
| L(s) = 1 | − 1.27·2-s − 0.0371·3-s + 0.619·4-s − 1.45·5-s + 0.0472·6-s + 0.927·7-s + 0.483·8-s − 0.998·9-s + 1.85·10-s + 0.301·11-s − 0.0230·12-s − 0.888·13-s − 1.18·14-s + 0.0540·15-s − 1.23·16-s + 0.907·17-s + 1.27·18-s − 1.56·19-s − 0.902·20-s − 0.0344·21-s − 0.383·22-s + 1.51·23-s − 0.0179·24-s + 1.12·25-s + 1.13·26-s + 0.0742·27-s + 0.575·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4227607789\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4227607789\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 131 | \( 1 + T \) |
| good | 2 | \( 1 + 1.79T + 2T^{2} \) |
| 3 | \( 1 + 0.0643T + 3T^{2} \) |
| 5 | \( 1 + 3.25T + 5T^{2} \) |
| 7 | \( 1 - 2.45T + 7T^{2} \) |
| 13 | \( 1 + 3.20T + 13T^{2} \) |
| 17 | \( 1 - 3.74T + 17T^{2} \) |
| 19 | \( 1 + 6.81T + 19T^{2} \) |
| 23 | \( 1 - 7.26T + 23T^{2} \) |
| 29 | \( 1 + 6.92T + 29T^{2} \) |
| 31 | \( 1 + 2.81T + 31T^{2} \) |
| 37 | \( 1 - 1.68T + 37T^{2} \) |
| 41 | \( 1 + 3.36T + 41T^{2} \) |
| 43 | \( 1 + 1.13T + 43T^{2} \) |
| 47 | \( 1 - 2.03T + 47T^{2} \) |
| 53 | \( 1 - 0.227T + 53T^{2} \) |
| 59 | \( 1 - 3.11T + 59T^{2} \) |
| 61 | \( 1 + 2.47T + 61T^{2} \) |
| 67 | \( 1 + 12.0T + 67T^{2} \) |
| 71 | \( 1 + 1.91T + 71T^{2} \) |
| 73 | \( 1 - 14.4T + 73T^{2} \) |
| 79 | \( 1 - 6.50T + 79T^{2} \) |
| 83 | \( 1 - 16.1T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 - 2.59T + 97T^{2} \) |
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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
−9.167591191843062041598808583379, −8.763040320330265308672517612587, −7.82970533827609180424883188972, −7.69070398924160041858493238453, −6.66770573890966108718668282052, −5.22933394559432296459802829366, −4.48602878786409316195830754307, −3.41144967923536134371717743962, −2.02398578137733778983413216475, −0.55460516641327928856392422644,
0.55460516641327928856392422644, 2.02398578137733778983413216475, 3.41144967923536134371717743962, 4.48602878786409316195830754307, 5.22933394559432296459802829366, 6.66770573890966108718668282052, 7.69070398924160041858493238453, 7.82970533827609180424883188972, 8.763040320330265308672517612587, 9.167591191843062041598808583379
