L(s) = 1 | − 5-s − 7-s − 6·11-s + 13-s + 19-s + 6·23-s + 25-s − 6·29-s + 8·31-s + 35-s + 7·37-s − 6·41-s + 4·43-s + 12·47-s − 6·49-s + 6·53-s + 6·55-s − 11·61-s − 65-s + 7·67-s − 6·71-s + 11·73-s + 6·77-s − 79-s − 6·83-s − 12·89-s − 91-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.377·7-s − 1.80·11-s + 0.277·13-s + 0.229·19-s + 1.25·23-s + 1/5·25-s − 1.11·29-s + 1.43·31-s + 0.169·35-s + 1.15·37-s − 0.937·41-s + 0.609·43-s + 1.75·47-s − 6/7·49-s + 0.824·53-s + 0.809·55-s − 1.40·61-s − 0.124·65-s + 0.855·67-s − 0.712·71-s + 1.28·73-s + 0.683·77-s − 0.112·79-s − 0.658·83-s − 1.27·89-s − 0.104·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−7.50162656269519084362976400291, −6.86507638848018767652206162448, −5.97729307041736132919623486313, −5.33507900212300617584171680729, −4.69113144553480439124762990414, −3.84153151061395448138263887916, −2.93168478103303074817675151239, −2.49067168687006530055039794426, −1.08590041868368686981822715362, 0,
1.08590041868368686981822715362, 2.49067168687006530055039794426, 2.93168478103303074817675151239, 3.84153151061395448138263887916, 4.69113144553480439124762990414, 5.33507900212300617584171680729, 5.97729307041736132919623486313, 6.86507638848018767652206162448, 7.50162656269519084362976400291