| L(s) = 1 | − 5-s − 7-s + 3·11-s − 5·13-s − 3·17-s + 2·19-s − 6·23-s + 25-s + 3·29-s + 4·31-s + 35-s − 2·37-s + 12·41-s − 10·43-s + 9·47-s + 49-s + 12·53-s − 3·55-s − 8·61-s + 5·65-s − 4·67-s + 2·73-s − 3·77-s + 79-s − 12·83-s + 3·85-s + 12·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s + 0.904·11-s − 1.38·13-s − 0.727·17-s + 0.458·19-s − 1.25·23-s + 1/5·25-s + 0.557·29-s + 0.718·31-s + 0.169·35-s − 0.328·37-s + 1.87·41-s − 1.52·43-s + 1.31·47-s + 1/7·49-s + 1.64·53-s − 0.404·55-s − 1.02·61-s + 0.620·65-s − 0.488·67-s + 0.234·73-s − 0.341·77-s + 0.112·79-s − 1.31·83-s + 0.325·85-s + 1.27·89-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 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−15.97715274404588, −15.36952797494398, −14.83945845380390, −14.34237364891917, −13.75495285587948, −13.30199006424507, −12.32734760676798, −12.17402968395493, −11.71587711143152, −10.99785736890617, −10.27329458834345, −9.841051868164652, −9.235255068576990, −8.691060868681023, −7.994573831200012, −7.337819282688609, −6.908930520909077, −6.210998942700925, −5.608636476454559, −4.667549310881372, −4.284778172470667, −3.559675132912979, −2.709076890803274, −2.097821018748656, −0.9697620591159515, 0,
0.9697620591159515, 2.097821018748656, 2.709076890803274, 3.559675132912979, 4.284778172470667, 4.667549310881372, 5.608636476454559, 6.210998942700925, 6.908930520909077, 7.337819282688609, 7.994573831200012, 8.691060868681023, 9.235255068576990, 9.841051868164652, 10.27329458834345, 10.99785736890617, 11.71587711143152, 12.17402968395493, 12.32734760676798, 13.30199006424507, 13.75495285587948, 14.34237364891917, 14.83945845380390, 15.36952797494398, 15.97715274404588
