| L(s) = 1 | − 3-s − 5-s + 7-s − 2·9-s + 11-s − 13-s + 15-s + 17-s − 19-s − 21-s + 23-s + 25-s + 5·27-s − 3·29-s + 10·31-s − 33-s − 35-s − 10·37-s + 39-s − 10·41-s + 6·43-s + 2·45-s + 8·47-s − 6·49-s − 51-s − 9·53-s − 55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s − 2/3·9-s + 0.301·11-s − 0.277·13-s + 0.258·15-s + 0.242·17-s − 0.229·19-s − 0.218·21-s + 0.208·23-s + 1/5·25-s + 0.962·27-s − 0.557·29-s + 1.79·31-s − 0.174·33-s − 0.169·35-s − 1.64·37-s + 0.160·39-s − 1.56·41-s + 0.914·43-s + 0.298·45-s + 1.16·47-s − 6/7·49-s − 0.140·51-s − 1.23·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 66880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−14.35358452110060, −14.07556501175920, −13.48199436030789, −12.84299896687531, −12.13219166627201, −12.03684029010699, −11.43955840768330, −11.04273844531685, −10.40541828423402, −10.08691726389609, −9.240713985419437, −8.796909942405852, −8.251394261009599, −7.823216923808194, −7.123093953371890, −6.589685687251028, −6.121989507875821, −5.413109392579422, −4.955232209673531, −4.478394150742676, −3.669524640672963, −3.145544446645895, −2.420426539615951, −1.613252716900733, −0.7930635790057354, 0,
0.7930635790057354, 1.613252716900733, 2.420426539615951, 3.145544446645895, 3.669524640672963, 4.478394150742676, 4.955232209673531, 5.413109392579422, 6.121989507875821, 6.589685687251028, 7.123093953371890, 7.823216923808194, 8.251394261009599, 8.796909942405852, 9.240713985419437, 10.08691726389609, 10.40541828423402, 11.04273844531685, 11.43955840768330, 12.03684029010699, 12.13219166627201, 12.84299896687531, 13.48199436030789, 14.07556501175920, 14.35358452110060
