| L(s) = 1 | − 1.56·3-s + 5-s + 1.56·7-s − 0.561·9-s − 2·13-s − 1.56·15-s − 3.56·17-s − 1.56·19-s − 2.43·21-s + 3.12·23-s + 25-s + 5.56·27-s + 2.68·29-s − 2.43·31-s + 1.56·35-s + 6.68·37-s + 3.12·39-s − 2·41-s − 6.24·43-s − 0.561·45-s + 4.87·47-s − 4.56·49-s + 5.56·51-s + 0.438·53-s + 2.43·57-s + 7.12·59-s − 14.6·61-s + ⋯ |
| L(s) = 1 | − 0.901·3-s + 0.447·5-s + 0.590·7-s − 0.187·9-s − 0.554·13-s − 0.403·15-s − 0.863·17-s − 0.358·19-s − 0.532·21-s + 0.651·23-s + 0.200·25-s + 1.07·27-s + 0.498·29-s − 0.437·31-s + 0.263·35-s + 1.09·37-s + 0.500·39-s − 0.312·41-s − 0.952·43-s − 0.0837·45-s + 0.711·47-s − 0.651·49-s + 0.778·51-s + 0.0602·53-s + 0.322·57-s + 0.927·59-s − 1.88·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 + 1.56T + 3T^{2} \) |
| 7 | \( 1 - 1.56T + 7T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 3.56T + 17T^{2} \) |
| 19 | \( 1 + 1.56T + 19T^{2} \) |
| 23 | \( 1 - 3.12T + 23T^{2} \) |
| 29 | \( 1 - 2.68T + 29T^{2} \) |
| 31 | \( 1 + 2.43T + 31T^{2} \) |
| 37 | \( 1 - 6.68T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 6.24T + 43T^{2} \) |
| 47 | \( 1 - 4.87T + 47T^{2} \) |
| 53 | \( 1 - 0.438T + 53T^{2} \) |
| 59 | \( 1 - 7.12T + 59T^{2} \) |
| 61 | \( 1 + 14.6T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 - 8.68T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 9.36T + 79T^{2} \) |
| 83 | \( 1 + 3.12T + 83T^{2} \) |
| 89 | \( 1 + 8.43T + 89T^{2} \) |
| 97 | \( 1 + 1.12T + 97T^{2} \) |
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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
−7.15673988308875618716612719000, −6.55095159732329623875788620217, −6.02190306012742267495219298326, −5.11389584672322255327558410707, −4.91117125486459745062827588908, −3.99970415302905559884714436781, −2.86166812998242893449730268559, −2.15524747745675525785596233658, −1.11865406624299135599817496595, 0,
1.11865406624299135599817496595, 2.15524747745675525785596233658, 2.86166812998242893449730268559, 3.99970415302905559884714436781, 4.91117125486459745062827588908, 5.11389584672322255327558410707, 6.02190306012742267495219298326, 6.55095159732329623875788620217, 7.15673988308875618716612719000
