L(s) = 1 | − 3-s − 5-s − 7-s − 2·9-s + 6·11-s − 3·13-s + 15-s − 2·17-s − 7·19-s + 21-s + 3·23-s − 4·25-s + 5·27-s − 4·31-s − 6·33-s + 35-s − 4·37-s + 3·39-s + 12·41-s + 2·45-s − 6·47-s + 49-s + 2·51-s − 12·53-s − 6·55-s + 7·57-s + 5·59-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s − 2/3·9-s + 1.80·11-s − 0.832·13-s + 0.258·15-s − 0.485·17-s − 1.60·19-s + 0.218·21-s + 0.625·23-s − 4/5·25-s + 0.962·27-s − 0.718·31-s − 1.04·33-s + 0.169·35-s − 0.657·37-s + 0.480·39-s + 1.87·41-s + 0.298·45-s − 0.875·47-s + 1/7·49-s + 0.280·51-s − 1.64·53-s − 0.809·55-s + 0.927·57-s + 0.650·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 29 | \( 1 \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 17 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 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
−14.66164173082551, −14.29882559150734, −14.19528460875573, −13.06161059254599, −12.63812029668429, −12.39661196007511, −11.55323981288939, −11.27760922648004, −11.04936028269042, −10.09348193550574, −9.664441377666329, −9.037193384630804, −8.617347813737643, −8.095397543842479, −7.107668487159998, −6.926964159954793, −6.172817987847168, −5.922947150956643, −5.021108935382073, −4.437546486765102, −3.911529560246128, −3.322737260332623, −2.429274014380739, −1.796976276664585, −0.7363225449312661, 0,
0.7363225449312661, 1.796976276664585, 2.429274014380739, 3.322737260332623, 3.911529560246128, 4.437546486765102, 5.021108935382073, 5.922947150956643, 6.172817987847168, 6.926964159954793, 7.107668487159998, 8.095397543842479, 8.617347813737643, 9.037193384630804, 9.664441377666329, 10.09348193550574, 11.04936028269042, 11.27760922648004, 11.55323981288939, 12.39661196007511, 12.63812029668429, 13.06161059254599, 14.19528460875573, 14.29882559150734, 14.66164173082551