| L(s) = 1 | − 2·5-s − 4·7-s + 11-s − 6·13-s + 2·17-s + 8·23-s − 25-s + 10·29-s + 8·31-s + 8·35-s − 37-s − 6·41-s + 8·47-s + 9·49-s + 6·53-s − 2·55-s − 4·59-s + 2·61-s + 12·65-s − 12·67-s + 2·73-s − 4·77-s − 4·79-s − 4·83-s − 4·85-s + 6·89-s + 24·91-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.51·7-s + 0.301·11-s − 1.66·13-s + 0.485·17-s + 1.66·23-s − 1/5·25-s + 1.85·29-s + 1.43·31-s + 1.35·35-s − 0.164·37-s − 0.937·41-s + 1.16·47-s + 9/7·49-s + 0.824·53-s − 0.269·55-s − 0.520·59-s + 0.256·61-s + 1.48·65-s − 1.46·67-s + 0.234·73-s − 0.455·77-s − 0.450·79-s − 0.439·83-s − 0.433·85-s + 0.635·89-s + 2.51·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 234432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.10047192251095, −12.45488334559165, −12.14617876829068, −12.05474895195040, −11.42869791241347, −10.72941349502854, −10.23842620348422, −9.913932954516849, −9.525628799200993, −8.906912331862689, −8.521211098226426, −7.905572110033013, −7.350563520443799, −6.967670001694896, −6.656158795745779, −6.056066962622555, −5.414468595053753, −4.808249718326780, −4.428669062103123, −3.825253544372547, −3.124611459598473, −2.877484579960220, −2.407116897005520, −1.258471271108034, −0.6533018591874171, 0,
0.6533018591874171, 1.258471271108034, 2.407116897005520, 2.877484579960220, 3.124611459598473, 3.825253544372547, 4.428669062103123, 4.808249718326780, 5.414468595053753, 6.056066962622555, 6.656158795745779, 6.967670001694896, 7.350563520443799, 7.905572110033013, 8.521211098226426, 8.906912331862689, 9.525628799200993, 9.913932954516849, 10.23842620348422, 10.72941349502854, 11.42869791241347, 12.05474895195040, 12.14617876829068, 12.45488334559165, 13.10047192251095
