| L(s) = 1 | − 2·3-s + 5-s + 7-s + 9-s − 11-s + 3·13-s − 2·15-s − 7·17-s − 2·21-s − 23-s + 25-s + 4·27-s − 29-s − 4·31-s + 2·33-s + 35-s + 4·37-s − 6·39-s + 5·41-s + 8·43-s + 45-s + 2·47-s + 49-s + 14·51-s + 4·53-s − 55-s + 5·59-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.832·13-s − 0.516·15-s − 1.69·17-s − 0.436·21-s − 0.208·23-s + 1/5·25-s + 0.769·27-s − 0.185·29-s − 0.718·31-s + 0.348·33-s + 0.169·35-s + 0.657·37-s − 0.960·39-s + 0.780·41-s + 1.21·43-s + 0.149·45-s + 0.291·47-s + 1/7·49-s + 1.96·51-s + 0.549·53-s − 0.134·55-s + 0.650·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 + 11 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.22960006974694, −12.84596368710635, −12.31470721379174, −11.76196667056693, −11.32865609808308, −10.96406658890750, −10.64549127817637, −10.26195841609479, −9.402828017373324, −9.129591859115609, −8.572974078213031, −8.137838038482099, −7.372443290204807, −7.001087234950906, −6.403867389921818, −5.868737163687948, −5.744612768371754, −5.075946566876566, −4.406807125047198, −4.233201693607921, −3.369474776970746, −2.562841602265312, −2.158183964594018, −1.378099399229301, −0.7349952187574182, 0,
0.7349952187574182, 1.378099399229301, 2.158183964594018, 2.562841602265312, 3.369474776970746, 4.233201693607921, 4.406807125047198, 5.075946566876566, 5.744612768371754, 5.868737163687948, 6.403867389921818, 7.001087234950906, 7.372443290204807, 8.137838038482099, 8.572974078213031, 9.129591859115609, 9.402828017373324, 10.26195841609479, 10.64549127817637, 10.96406658890750, 11.32865609808308, 11.76196667056693, 12.31470721379174, 12.84596368710635, 13.22960006974694
