L(s) = 1 | − 2-s − 2·3-s + 4-s + 3·5-s + 2·6-s − 8-s + 9-s − 3·10-s − 2·12-s − 2·13-s − 6·15-s + 16-s − 18-s + 7·19-s + 3·20-s + 3·23-s + 2·24-s + 4·25-s + 2·26-s + 4·27-s + 6·29-s + 6·30-s − 4·31-s − 32-s + 36-s + 7·37-s − 7·38-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 1.34·5-s + 0.816·6-s − 0.353·8-s + 1/3·9-s − 0.948·10-s − 0.577·12-s − 0.554·13-s − 1.54·15-s + 1/4·16-s − 0.235·18-s + 1.60·19-s + 0.670·20-s + 0.625·23-s + 0.408·24-s + 4/5·25-s + 0.392·26-s + 0.769·27-s + 1.11·29-s + 1.09·30-s − 0.718·31-s − 0.176·32-s + 1/6·36-s + 1.15·37-s − 1.13·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.574102023\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.574102023\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 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
−15.46628652901989, −14.50112768706236, −14.25053209974000, −13.64023692836489, −12.91489560821007, −12.47749288542635, −11.89614982490113, −11.26715275488390, −10.98196087983875, −10.22087241457853, −9.870103284093227, −9.340496241073269, −8.931115046169017, −7.946271818568786, −7.504800291435339, −6.634943047242674, −6.375519627583465, −5.715481971314339, −5.136336452597307, −4.852898998071724, −3.587033289525616, −2.734184918496216, −2.172326362079915, −1.163236509140706, −0.6791918632346270,
0.6791918632346270, 1.163236509140706, 2.172326362079915, 2.734184918496216, 3.587033289525616, 4.852898998071724, 5.136336452597307, 5.715481971314339, 6.375519627583465, 6.634943047242674, 7.504800291435339, 7.946271818568786, 8.931115046169017, 9.340496241073269, 9.870103284093227, 10.22087241457853, 10.98196087983875, 11.26715275488390, 11.89614982490113, 12.47749288542635, 12.91489560821007, 13.64023692836489, 14.25053209974000, 14.50112768706236, 15.46628652901989