L(s) = 1 | − 3-s + 2·5-s − 7-s + 9-s + 4·11-s − 2·15-s − 6·17-s + 4·19-s + 21-s − 25-s − 27-s − 2·29-s − 4·33-s − 2·35-s − 6·37-s − 2·41-s + 4·43-s + 2·45-s + 49-s + 6·51-s + 6·53-s + 8·55-s − 4·57-s + 12·59-s − 2·61-s − 63-s + 4·67-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.516·15-s − 1.45·17-s + 0.917·19-s + 0.218·21-s − 1/5·25-s − 0.192·27-s − 0.371·29-s − 0.696·33-s − 0.338·35-s − 0.986·37-s − 0.312·41-s + 0.609·43-s + 0.298·45-s + 1/7·49-s + 0.840·51-s + 0.824·53-s + 1.07·55-s − 0.529·57-s + 1.56·59-s − 0.256·61-s − 0.125·63-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.120052812\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.120052812\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−14.19493311424709, −13.87931195348763, −13.31332729001332, −12.99642191193333, −12.22730295699042, −11.85926171323180, −11.35016266653692, −10.81161621060825, −10.27660565409717, −9.624406687771320, −9.380193480252302, −8.829590511888168, −8.246162784263372, −7.267319768172702, −6.960430832981696, −6.358427145275523, −6.005155713970808, −5.304338158620820, −4.873762008765878, −3.929544645840822, −3.706643760984163, −2.630019083054870, −2.026787675470422, −1.365323585679590, −0.5332794516868203,
0.5332794516868203, 1.365323585679590, 2.026787675470422, 2.630019083054870, 3.706643760984163, 3.929544645840822, 4.873762008765878, 5.304338158620820, 6.005155713970808, 6.358427145275523, 6.960430832981696, 7.267319768172702, 8.246162784263372, 8.829590511888168, 9.380193480252302, 9.624406687771320, 10.27660565409717, 10.81161621060825, 11.35016266653692, 11.85926171323180, 12.22730295699042, 12.99642191193333, 13.31332729001332, 13.87931195348763, 14.19493311424709