| L(s) = 1 | − 5-s − 7-s − 4·11-s − 2·13-s + 2·17-s + 4·23-s + 25-s + 2·29-s + 8·31-s + 35-s − 10·37-s − 2·41-s − 4·43-s − 4·47-s + 49-s + 10·53-s + 4·55-s + 4·59-s + 2·61-s + 2·65-s − 4·67-s − 6·73-s + 4·77-s + 8·79-s − 4·83-s − 2·85-s + 6·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s − 1.20·11-s − 0.554·13-s + 0.485·17-s + 0.834·23-s + 1/5·25-s + 0.371·29-s + 1.43·31-s + 0.169·35-s − 1.64·37-s − 0.312·41-s − 0.609·43-s − 0.583·47-s + 1/7·49-s + 1.37·53-s + 0.539·55-s + 0.520·59-s + 0.256·61-s + 0.248·65-s − 0.488·67-s − 0.702·73-s + 0.455·77-s + 0.900·79-s − 0.439·83-s − 0.216·85-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−15.81254717380869, −15.46511113013627, −14.93545961581922, −14.34781121407570, −13.61908667132711, −13.22299607366886, −12.67207867107638, −11.95730891434205, −11.77142089889209, −10.84539179950388, −10.24977571290394, −10.07642635297247, −9.188092288351481, −8.527437784100698, −8.078439411552757, −7.385965758122611, −6.914924179888457, −6.245768861278075, −5.295041582688088, −5.046407976633177, −4.235388360984413, −3.328753162675835, −2.889981345425118, −2.087703396369281, −0.9310352554922978, 0,
0.9310352554922978, 2.087703396369281, 2.889981345425118, 3.328753162675835, 4.235388360984413, 5.046407976633177, 5.295041582688088, 6.245768861278075, 6.914924179888457, 7.385965758122611, 8.078439411552757, 8.527437784100698, 9.188092288351481, 10.07642635297247, 10.24977571290394, 10.84539179950388, 11.77142089889209, 11.95730891434205, 12.67207867107638, 13.22299607366886, 13.61908667132711, 14.34781121407570, 14.93545961581922, 15.46511113013627, 15.81254717380869
