| L(s) = 1 | − 3-s + 5-s + 4·7-s + 9-s − 13-s − 15-s + 17-s − 4·19-s − 4·21-s − 8·23-s + 25-s − 27-s + 10·29-s − 4·31-s + 4·35-s − 10·37-s + 39-s − 10·41-s + 4·43-s + 45-s + 8·47-s + 9·49-s − 51-s + 10·53-s + 4·57-s − 4·59-s + 2·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s − 0.277·13-s − 0.258·15-s + 0.242·17-s − 0.917·19-s − 0.872·21-s − 1.66·23-s + 1/5·25-s − 0.192·27-s + 1.85·29-s − 0.718·31-s + 0.676·35-s − 1.64·37-s + 0.160·39-s − 1.56·41-s + 0.609·43-s + 0.149·45-s + 1.16·47-s + 9/7·49-s − 0.140·51-s + 1.37·53-s + 0.529·57-s − 0.520·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 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
−13.34877707718960, −12.55697494793922, −12.19760821200812, −11.88129159923075, −11.50145570047629, −10.72942879814123, −10.46639812588576, −10.24371194007626, −9.566513275770092, −8.871709281461073, −8.446527286704122, −8.133177586942791, −7.548509395854097, −6.910657070077069, −6.606404237441091, −5.865681891070748, −5.436519268149885, −5.112632794220799, −4.360230081619024, −4.198511462995224, −3.388344224421061, −2.510166165415773, −1.990869435199573, −1.595392428116956, −0.8537040928009451, 0,
0.8537040928009451, 1.595392428116956, 1.990869435199573, 2.510166165415773, 3.388344224421061, 4.198511462995224, 4.360230081619024, 5.112632794220799, 5.436519268149885, 5.865681891070748, 6.606404237441091, 6.910657070077069, 7.548509395854097, 8.133177586942791, 8.446527286704122, 8.871709281461073, 9.566513275770092, 10.24371194007626, 10.46639812588576, 10.72942879814123, 11.50145570047629, 11.88129159923075, 12.19760821200812, 12.55697494793922, 13.34877707718960
