| L(s) = 1 | − 4·3-s + 5·5-s + 7·7-s − 11·9-s + 20·11-s − 10·13-s − 20·15-s − 14·17-s + 12·19-s − 28·21-s + 104·23-s + 25·25-s + 152·27-s − 122·29-s + 224·31-s − 80·33-s + 35·35-s + 158·37-s + 40·39-s + 378·41-s + 404·43-s − 55·45-s + 112·47-s + 49·49-s + 56·51-s + 270·53-s + 100·55-s + ⋯ |
| L(s) = 1 | − 0.769·3-s + 0.447·5-s + 0.377·7-s − 0.407·9-s + 0.548·11-s − 0.213·13-s − 0.344·15-s − 0.199·17-s + 0.144·19-s − 0.290·21-s + 0.942·23-s + 1/5·25-s + 1.08·27-s − 0.781·29-s + 1.29·31-s − 0.422·33-s + 0.169·35-s + 0.702·37-s + 0.164·39-s + 1.43·41-s + 1.43·43-s − 0.182·45-s + 0.347·47-s + 1/7·49-s + 0.153·51-s + 0.699·53-s + 0.245·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.504906826\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.504906826\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - p T \) |
| 7 | \( 1 - p T \) |
| good | 3 | \( 1 + 4 T + p^{3} T^{2} \) |
| 11 | \( 1 - 20 T + p^{3} T^{2} \) |
| 13 | \( 1 + 10 T + p^{3} T^{2} \) |
| 17 | \( 1 + 14 T + p^{3} T^{2} \) |
| 19 | \( 1 - 12 T + p^{3} T^{2} \) |
| 23 | \( 1 - 104 T + p^{3} T^{2} \) |
| 29 | \( 1 + 122 T + p^{3} T^{2} \) |
| 31 | \( 1 - 224 T + p^{3} T^{2} \) |
| 37 | \( 1 - 158 T + p^{3} T^{2} \) |
| 41 | \( 1 - 378 T + p^{3} T^{2} \) |
| 43 | \( 1 - 404 T + p^{3} T^{2} \) |
| 47 | \( 1 - 112 T + p^{3} T^{2} \) |
| 53 | \( 1 - 270 T + p^{3} T^{2} \) |
| 59 | \( 1 - 324 T + p^{3} T^{2} \) |
| 61 | \( 1 + 186 T + p^{3} T^{2} \) |
| 67 | \( 1 - 156 T + p^{3} T^{2} \) |
| 71 | \( 1 + 360 T + p^{3} T^{2} \) |
| 73 | \( 1 + 102 T + p^{3} T^{2} \) |
| 79 | \( 1 + 912 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1068 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1590 T + p^{3} T^{2} \) |
| 97 | \( 1 - 866 T + p^{3} T^{2} \) |
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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
−11.36982150837933341472128785434, −10.70309307718671433772954239193, −9.539466261997854894702472730654, −8.662685696865182574105792327692, −7.38451808224180478292108614141, −6.27274078789343100847431864859, −5.46095044432320784680238574967, −4.34987915649226139596483669709, −2.63299502056220013954943587164, −0.930725762682141429762899613916,
0.930725762682141429762899613916, 2.63299502056220013954943587164, 4.34987915649226139596483669709, 5.46095044432320784680238574967, 6.27274078789343100847431864859, 7.38451808224180478292108614141, 8.662685696865182574105792327692, 9.539466261997854894702472730654, 10.70309307718671433772954239193, 11.36982150837933341472128785434
