| L(s) = 1 | − 3-s − 5-s + 3·7-s + 9-s + 3·11-s − 13-s + 15-s − 17-s + 19-s − 3·21-s − 6·23-s + 25-s − 27-s + 29-s − 2·31-s − 3·33-s − 3·35-s + 3·37-s + 39-s + 7·41-s − 2·43-s − 45-s + 9·47-s + 2·49-s + 51-s − 13·53-s − 3·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.13·7-s + 1/3·9-s + 0.904·11-s − 0.277·13-s + 0.258·15-s − 0.242·17-s + 0.229·19-s − 0.654·21-s − 1.25·23-s + 1/5·25-s − 0.192·27-s + 0.185·29-s − 0.359·31-s − 0.522·33-s − 0.507·35-s + 0.493·37-s + 0.160·39-s + 1.09·41-s − 0.304·43-s − 0.149·45-s + 1.31·47-s + 2/7·49-s + 0.140·51-s − 1.78·53-s − 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 13 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−15.74879207251393, −14.83856306102290, −14.60545297897679, −14.06339558821420, −13.51096301921204, −12.66313042690021, −12.25856818134172, −11.71497323209917, −11.31263469372043, −10.90885175260955, −10.17629517280327, −9.629844724450283, −8.914719989442048, −8.427788117105285, −7.571305214336916, −7.497269625913992, −6.532565005486798, −6.017331758623450, −5.388292973732997, −4.568593069520553, −4.307559738115594, −3.581671396142058, −2.574250151396903, −1.728465524844939, −1.090918285940885, 0,
1.090918285940885, 1.728465524844939, 2.574250151396903, 3.581671396142058, 4.307559738115594, 4.568593069520553, 5.388292973732997, 6.017331758623450, 6.532565005486798, 7.497269625913992, 7.571305214336916, 8.427788117105285, 8.914719989442048, 9.629844724450283, 10.17629517280327, 10.90885175260955, 11.31263469372043, 11.71497323209917, 12.25856818134172, 12.66313042690021, 13.51096301921204, 14.06339558821420, 14.60545297897679, 14.83856306102290, 15.74879207251393
