| L(s) = 1 | + 3-s + 4·5-s + 9-s + 4·11-s + 13-s + 4·15-s − 17-s − 19-s − 2·23-s + 11·25-s + 27-s − 6·29-s − 9·31-s + 4·33-s + 11·37-s + 39-s − 10·41-s − 7·43-s + 4·45-s + 6·47-s − 51-s + 6·53-s + 16·55-s − 57-s − 8·59-s + 6·61-s + 4·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.78·5-s + 1/3·9-s + 1.20·11-s + 0.277·13-s + 1.03·15-s − 0.242·17-s − 0.229·19-s − 0.417·23-s + 11/5·25-s + 0.192·27-s − 1.11·29-s − 1.61·31-s + 0.696·33-s + 1.80·37-s + 0.160·39-s − 1.56·41-s − 1.06·43-s + 0.596·45-s + 0.875·47-s − 0.140·51-s + 0.824·53-s + 2.15·55-s − 0.132·57-s − 1.04·59-s + 0.768·61-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.899964463\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.899964463\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−13.26157371196126, −13.00285277520804, −12.53539306913016, −11.85758275934577, −11.32093387329425, −10.83623203404967, −10.30188336661287, −9.785402525209232, −9.390301751656725, −9.128600181108334, −8.628832060146871, −8.080260150645162, −7.353222289050569, −6.809344655888308, −6.439591267753823, −5.926563632254836, −5.433366389099498, −4.948502807391505, −4.062805000730461, −3.786162033387794, −3.013836384977118, −2.380016874323286, −1.727940679314329, −1.611800985515567, −0.6503302138534915,
0.6503302138534915, 1.611800985515567, 1.727940679314329, 2.380016874323286, 3.013836384977118, 3.786162033387794, 4.062805000730461, 4.948502807391505, 5.433366389099498, 5.926563632254836, 6.439591267753823, 6.809344655888308, 7.353222289050569, 8.080260150645162, 8.628832060146871, 9.128600181108334, 9.390301751656725, 9.785402525209232, 10.30188336661287, 10.83623203404967, 11.32093387329425, 11.85758275934577, 12.53539306913016, 13.00285277520804, 13.26157371196126
