L(s) = 1 | + 3-s + 5-s + 7-s + 9-s − 11-s − 4·13-s + 15-s + 3·17-s − 7·19-s + 21-s − 9·23-s + 25-s + 27-s − 3·29-s + 2·31-s − 33-s + 35-s − 4·37-s − 4·39-s − 6·41-s − 43-s + 45-s − 6·47-s + 49-s + 3·51-s + 3·53-s − 55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.301·11-s − 1.10·13-s + 0.258·15-s + 0.727·17-s − 1.60·19-s + 0.218·21-s − 1.87·23-s + 1/5·25-s + 0.192·27-s − 0.557·29-s + 0.359·31-s − 0.174·33-s + 0.169·35-s − 0.657·37-s − 0.640·39-s − 0.937·41-s − 0.152·43-s + 0.149·45-s − 0.875·47-s + 1/7·49-s + 0.420·51-s + 0.412·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−8.039002102944050217419158781442, −7.37211116820457318429849067419, −6.51565947196226256965842246295, −5.77109913704476912791258892984, −4.91514209823538841953237959317, −4.22303767958713950276952657460, −3.27425433252608589333040764406, −2.26103534921873330185998765674, −1.74356417487452812542295634571, 0,
1.74356417487452812542295634571, 2.26103534921873330185998765674, 3.27425433252608589333040764406, 4.22303767958713950276952657460, 4.91514209823538841953237959317, 5.77109913704476912791258892984, 6.51565947196226256965842246295, 7.37211116820457318429849067419, 8.039002102944050217419158781442