| L(s) = 1 | + 5-s − 7-s − 4·11-s + 2·13-s − 2·17-s + 4·19-s + 25-s − 10·29-s − 35-s − 6·37-s + 6·41-s + 4·43-s + 8·47-s + 49-s + 6·53-s − 4·55-s − 4·59-s + 10·61-s + 2·65-s − 4·67-s + 16·71-s − 14·73-s + 4·77-s + 8·79-s − 4·83-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 + 1/5·25-s − 1.85·29-s − 0.169·35-s − 0.986·37-s + 0.937·41-s + 0.609·43-s + 1.16·47-s + 1/7·49-s + 0.824·53-s − 0.539·55-s − 0.520·59-s + 1.28·61-s + 0.248·65-s − 0.488·67-s + 1.89·71-s − 1.63·73-s + 0.455·77-s + 0.900·79-s − 0.439·83-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 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 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 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−15.89829733163510, −15.49272618612713, −14.90809747250576, −14.17646365048238, −13.64993585194176, −13.25060427980480, −12.74973709958697, −12.21841772604007, −11.37456963806839, −10.94910921635593, −10.40123204263078, −9.822373841457224, −9.201378507070524, −8.775637618634795, −7.958391814068338, −7.398309850362715, −6.907495537426728, −5.993878485809680, −5.577391140708633, −5.076955491209480, −4.101375559516929, −3.508063895085718, −2.662716408654380, −2.103877406876728, −1.071406972741805, 0,
1.071406972741805, 2.103877406876728, 2.662716408654380, 3.508063895085718, 4.101375559516929, 5.076955491209480, 5.577391140708633, 5.993878485809680, 6.907495537426728, 7.398309850362715, 7.958391814068338, 8.775637618634795, 9.201378507070524, 9.822373841457224, 10.40123204263078, 10.94910921635593, 11.37456963806839, 12.21841772604007, 12.74973709958697, 13.25060427980480, 13.64993585194176, 14.17646365048238, 14.90809747250576, 15.49272618612713, 15.89829733163510
