| L(s) = 1 | + 1.56·3-s − 5-s − 3.56·7-s − 0.561·9-s − 11-s − 3.12·13-s − 1.56·15-s + 5.56·17-s − 2.43·19-s − 5.56·21-s − 7.12·23-s + 25-s − 5.56·27-s − 0.438·29-s − 8.68·31-s − 1.56·33-s + 3.56·35-s + 9.80·37-s − 4.87·39-s − 10·41-s − 5.12·43-s + 0.561·45-s + 7.12·47-s + 5.68·49-s + 8.68·51-s − 4.43·53-s + 55-s + ⋯ |
| L(s) = 1 | + 0.901·3-s − 0.447·5-s − 1.34·7-s − 0.187·9-s − 0.301·11-s − 0.866·13-s − 0.403·15-s + 1.34·17-s − 0.559·19-s − 1.21·21-s − 1.48·23-s + 0.200·25-s − 1.07·27-s − 0.0814·29-s − 1.55·31-s − 0.271·33-s + 0.602·35-s + 1.61·37-s − 0.780·39-s − 1.56·41-s − 0.781·43-s + 0.0837·45-s + 1.03·47-s + 0.812·49-s + 1.21·51-s − 0.609·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 1.56T + 3T^{2} \) |
| 7 | \( 1 + 3.56T + 7T^{2} \) |
| 13 | \( 1 + 3.12T + 13T^{2} \) |
| 17 | \( 1 - 5.56T + 17T^{2} \) |
| 19 | \( 1 + 2.43T + 19T^{2} \) |
| 23 | \( 1 + 7.12T + 23T^{2} \) |
| 29 | \( 1 + 0.438T + 29T^{2} \) |
| 31 | \( 1 + 8.68T + 31T^{2} \) |
| 37 | \( 1 - 9.80T + 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 + 5.12T + 43T^{2} \) |
| 47 | \( 1 - 7.12T + 47T^{2} \) |
| 53 | \( 1 + 4.43T + 53T^{2} \) |
| 59 | \( 1 - 13.3T + 59T^{2} \) |
| 61 | \( 1 + 3.56T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 2.43T + 71T^{2} \) |
| 73 | \( 1 - 4.87T + 73T^{2} \) |
| 79 | \( 1 + 0.876T + 79T^{2} \) |
| 83 | \( 1 + 10T + 83T^{2} \) |
| 89 | \( 1 - 9.80T + 89T^{2} \) |
| 97 | \( 1 - 17.1T + 97T^{2} \) |
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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
−9.774162422851143148989143093806, −8.893630754871880444806540824586, −7.989753250588789394736262770162, −7.39710619546079298511168971894, −6.29362391670958574231626920209, −5.36972290565777573677137287941, −3.90544820602947799556781224964, −3.24526656552310273598178079111, −2.26447301577735049107188602743, 0,
2.26447301577735049107188602743, 3.24526656552310273598178079111, 3.90544820602947799556781224964, 5.36972290565777573677137287941, 6.29362391670958574231626920209, 7.39710619546079298511168971894, 7.989753250588789394736262770162, 8.893630754871880444806540824586, 9.774162422851143148989143093806
