| L(s) = 1 | + 5-s + 4·7-s + 2·11-s − 4·13-s + 17-s − 5·19-s − 5·23-s + 25-s − 8·29-s − 7·31-s + 4·35-s + 6·37-s + 6·41-s − 2·43-s − 8·47-s + 9·49-s − 9·53-s + 2·55-s + 4·59-s − 13·61-s − 4·65-s − 10·67-s + 6·71-s − 6·73-s + 8·77-s − 9·79-s − 17·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.51·7-s + 0.603·11-s − 1.10·13-s + 0.242·17-s − 1.14·19-s − 1.04·23-s + 1/5·25-s − 1.48·29-s − 1.25·31-s + 0.676·35-s + 0.986·37-s + 0.937·41-s − 0.304·43-s − 1.16·47-s + 9/7·49-s − 1.23·53-s + 0.269·55-s + 0.520·59-s − 1.66·61-s − 0.496·65-s − 1.22·67-s + 0.712·71-s − 0.702·73-s + 0.911·77-s − 1.01·79-s − 1.86·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 + 17 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−7.56883127737305415311393528032, −6.80679679813662430749338369583, −5.88504425381693499042970167636, −5.44058353575193341819534685132, −4.46721073191146950941766920681, −4.20131967478516429729407971743, −2.95056694733365579276034112981, −1.90677667970264271775360494434, −1.61725955093687237692772250139, 0,
1.61725955093687237692772250139, 1.90677667970264271775360494434, 2.95056694733365579276034112981, 4.20131967478516429729407971743, 4.46721073191146950941766920681, 5.44058353575193341819534685132, 5.88504425381693499042970167636, 6.80679679813662430749338369583, 7.56883127737305415311393528032
