| L(s) = 1 | + 2-s − 3·3-s − 3·4-s + 5·5-s − 3·6-s − 7·8-s + 9·9-s + 5·10-s + 9·12-s − 15·15-s + 5·16-s − 14·17-s + 9·18-s − 22·19-s − 15·20-s + 34·23-s + 21·24-s + 25·25-s − 27·27-s − 15·30-s + 2·31-s + 33·32-s − 14·34-s − 27·36-s − 22·38-s − 35·40-s + 45·45-s + ⋯ |
| L(s) = 1 | + 1/2·2-s − 3-s − 3/4·4-s + 5-s − 1/2·6-s − 7/8·8-s + 9-s + 1/2·10-s + 3/4·12-s − 15-s + 5/16·16-s − 0.823·17-s + 1/2·18-s − 1.15·19-s − 3/4·20-s + 1.47·23-s + 7/8·24-s + 25-s − 27-s − 1/2·30-s + 2/31·31-s + 1.03·32-s − 0.411·34-s − 3/4·36-s − 0.578·38-s − 7/8·40-s + 45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7476443612\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7476443612\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p T \) |
| 5 | \( 1 - p T \) |
| good | 2 | \( 1 - T + p^{2} T^{2} \) |
| 7 | \( ( 1 - p T )( 1 + p T ) \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( ( 1 - p T )( 1 + p T ) \) |
| 17 | \( 1 + 14 T + p^{2} T^{2} \) |
| 19 | \( 1 + 22 T + p^{2} T^{2} \) |
| 23 | \( 1 - 34 T + p^{2} T^{2} \) |
| 29 | \( ( 1 - p T )( 1 + p T ) \) |
| 31 | \( 1 - 2 T + p^{2} T^{2} \) |
| 37 | \( ( 1 - p T )( 1 + p T ) \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( ( 1 - p T )( 1 + p T ) \) |
| 47 | \( 1 + 14 T + p^{2} T^{2} \) |
| 53 | \( 1 + 86 T + p^{2} T^{2} \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( 1 + 118 T + p^{2} T^{2} \) |
| 67 | \( ( 1 - p T )( 1 + p T ) \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( ( 1 - p T )( 1 + p T ) \) |
| 79 | \( 1 - 98 T + p^{2} T^{2} \) |
| 83 | \( 1 - 154 T + p^{2} T^{2} \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( ( 1 - p T )( 1 + p T ) \) |
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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
−18.84216747882218125052134749858, −17.73632491044169124692694506319, −16.94133112513282712743435662192, −15.07325242580970539835333354209, −13.50572692666400849793767943581, −12.59137691919670652128806918476, −10.74164239244868709362486626799, −9.195310302182541695476289542121, −6.33181250203772271694008358554, −4.84192581422996258804553371125,
4.84192581422996258804553371125, 6.33181250203772271694008358554, 9.195310302182541695476289542121, 10.74164239244868709362486626799, 12.59137691919670652128806918476, 13.50572692666400849793767943581, 15.07325242580970539835333354209, 16.94133112513282712743435662192, 17.73632491044169124692694506319, 18.84216747882218125052134749858
