| L(s) = 1 | + 5-s + 7-s + 4·11-s + 2·13-s − 2·17-s + 4·19-s − 8·23-s + 25-s − 2·29-s + 35-s − 6·37-s + 6·41-s − 4·43-s + 49-s − 10·53-s + 4·55-s − 12·59-s − 14·61-s + 2·65-s − 12·67-s − 8·71-s + 10·73-s + 4·77-s − 16·79-s + 12·83-s − 2·85-s − 10·89-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s + 1.20·11-s + 0.554·13-s − 0.485·17-s + 0.917·19-s − 1.66·23-s + 1/5·25-s − 0.371·29-s + 0.169·35-s − 0.986·37-s + 0.937·41-s − 0.609·43-s + 1/7·49-s − 1.37·53-s + 0.539·55-s − 1.56·59-s − 1.79·61-s + 0.248·65-s − 1.46·67-s − 0.949·71-s + 1.17·73-s + 0.455·77-s − 1.80·79-s + 1.31·83-s − 0.216·85-s − 1.05·89-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 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−15.82777644969353, −15.52497904572033, −14.64239887249415, −14.29264457322113, −13.74050757664380, −13.44509873443981, −12.52545036881055, −12.03414777306236, −11.62189469836404, −10.92823702366567, −10.45958958421482, −9.689974716585640, −9.220677920998896, −8.787780269994214, −7.959682485402171, −7.523656557434341, −6.660711972286748, −6.166388652172635, −5.687418978893860, −4.811517515110441, −4.203940507305483, −3.556039358419806, −2.777809142554225, −1.662799210006684, −1.429566330321504, 0,
1.429566330321504, 1.662799210006684, 2.777809142554225, 3.556039358419806, 4.203940507305483, 4.811517515110441, 5.687418978893860, 6.166388652172635, 6.660711972286748, 7.523656557434341, 7.959682485402171, 8.787780269994214, 9.220677920998896, 9.689974716585640, 10.45958958421482, 10.92823702366567, 11.62189469836404, 12.03414777306236, 12.52545036881055, 13.44509873443981, 13.74050757664380, 14.29264457322113, 14.64239887249415, 15.52497904572033, 15.82777644969353
