| L(s) = 1 | − 3-s + 2·7-s − 2·9-s − 5·11-s + 5·17-s − 5·19-s − 2·21-s − 6·23-s + 5·27-s + 4·29-s − 10·31-s + 5·33-s − 10·37-s + 5·41-s − 4·43-s + 8·47-s − 3·49-s − 5·51-s − 10·53-s + 5·57-s − 10·61-s − 4·63-s − 3·67-s + 6·69-s − 5·73-s − 10·77-s − 10·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755·7-s − 2/3·9-s − 1.50·11-s + 1.21·17-s − 1.14·19-s − 0.436·21-s − 1.25·23-s + 0.962·27-s + 0.742·29-s − 1.79·31-s + 0.870·33-s − 1.64·37-s + 0.780·41-s − 0.609·43-s + 1.16·47-s − 3/7·49-s − 0.700·51-s − 1.37·53-s + 0.662·57-s − 1.28·61-s − 0.503·63-s − 0.366·67-s + 0.722·69-s − 0.585·73-s − 1.13·77-s − 1.12·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 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
−10.19247353139393510497215125211, −8.830850455477701050559569986904, −8.094469795338918061675120518148, −7.41230004297178578617275577832, −6.05631994479789712851621696197, −5.45104861185949508134672954635, −4.61867723730644924904633108854, −3.21115827901964811624216839347, −1.93887536017233006617656215932, 0,
1.93887536017233006617656215932, 3.21115827901964811624216839347, 4.61867723730644924904633108854, 5.45104861185949508134672954635, 6.05631994479789712851621696197, 7.41230004297178578617275577832, 8.094469795338918061675120518148, 8.830850455477701050559569986904, 10.19247353139393510497215125211
