| L(s) = 1 | − 3-s − 4·5-s + 2·7-s + 9-s − 11-s − 13-s + 4·15-s − 17-s + 4·19-s − 2·21-s + 6·23-s + 11·25-s − 27-s − 10·31-s + 33-s − 8·35-s + 12·37-s + 39-s + 2·41-s + 12·43-s − 4·45-s + 4·47-s − 3·49-s + 51-s + 6·53-s + 4·55-s − 4·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.78·5-s + 0.755·7-s + 1/3·9-s − 0.301·11-s − 0.277·13-s + 1.03·15-s − 0.242·17-s + 0.917·19-s − 0.436·21-s + 1.25·23-s + 11/5·25-s − 0.192·27-s − 1.79·31-s + 0.174·33-s − 1.35·35-s + 1.97·37-s + 0.160·39-s + 0.312·41-s + 1.82·43-s − 0.596·45-s + 0.583·47-s − 3/7·49-s + 0.140·51-s + 0.824·53-s + 0.539·55-s − 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116688 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.369021763\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.369021763\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.50287212416389, −12.79394126321341, −12.71042478815839, −11.94983788187992, −11.64172074712897, −11.24837518100675, −10.74640440587662, −10.61487656563192, −9.586308329668760, −9.069015269931264, −8.746163487917901, −7.807221481089245, −7.658382771938879, −7.358021802852553, −6.778671823979842, −5.925100069007521, −5.443982689284845, −4.868109108435137, −4.259333198326381, −4.147471438810666, −3.131729496067286, −2.848587927351849, −1.800064893721629, −0.9683338832841065, −0.4700358129934431,
0.4700358129934431, 0.9683338832841065, 1.800064893721629, 2.848587927351849, 3.131729496067286, 4.147471438810666, 4.259333198326381, 4.868109108435137, 5.443982689284845, 5.925100069007521, 6.778671823979842, 7.358021802852553, 7.658382771938879, 7.807221481089245, 8.746163487917901, 9.069015269931264, 9.586308329668760, 10.61487656563192, 10.74640440587662, 11.24837518100675, 11.64172074712897, 11.94983788187992, 12.71042478815839, 12.79394126321341, 13.50287212416389
