| L(s) = 1 | + 0.456·2-s + 3-s − 1.79·4-s − 5-s + 0.456·6-s + 4.37·7-s − 1.73·8-s + 9-s − 0.456·10-s − 1.79·12-s + 0.913·13-s + 1.99·14-s − 15-s + 2.79·16-s + 1.73·17-s + 0.456·18-s − 3.46·19-s + 1.79·20-s + 4.37·21-s − 0.582·23-s − 1.73·24-s + 25-s + 0.417·26-s + 27-s − 7.84·28-s + 9.66·29-s − 0.456·30-s + ⋯ |
| L(s) = 1 | + 0.323·2-s + 0.577·3-s − 0.895·4-s − 0.447·5-s + 0.186·6-s + 1.65·7-s − 0.612·8-s + 0.333·9-s − 0.144·10-s − 0.517·12-s + 0.253·13-s + 0.534·14-s − 0.258·15-s + 0.697·16-s + 0.420·17-s + 0.107·18-s − 0.794·19-s + 0.400·20-s + 0.955·21-s − 0.121·23-s − 0.353·24-s + 0.200·25-s + 0.0818·26-s + 0.192·27-s − 1.48·28-s + 1.79·29-s − 0.0834·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.266042083\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.266042083\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.456T + 2T^{2} \) |
| 7 | \( 1 - 4.37T + 7T^{2} \) |
| 13 | \( 1 - 0.913T + 13T^{2} \) |
| 17 | \( 1 - 1.73T + 17T^{2} \) |
| 19 | \( 1 + 3.46T + 19T^{2} \) |
| 23 | \( 1 + 0.582T + 23T^{2} \) |
| 29 | \( 1 - 9.66T + 29T^{2} \) |
| 31 | \( 1 + 8.58T + 31T^{2} \) |
| 37 | \( 1 - 7.58T + 37T^{2} \) |
| 41 | \( 1 - 3.46T + 41T^{2} \) |
| 43 | \( 1 + 9.66T + 43T^{2} \) |
| 47 | \( 1 - 3.41T + 47T^{2} \) |
| 53 | \( 1 - 12.1T + 53T^{2} \) |
| 59 | \( 1 - 13.5T + 59T^{2} \) |
| 61 | \( 1 + 3.55T + 61T^{2} \) |
| 67 | \( 1 + 4.41T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 5.10T + 73T^{2} \) |
| 79 | \( 1 - 0.818T + 79T^{2} \) |
| 83 | \( 1 + 15.8T + 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 - 4.41T + 97T^{2} \) |
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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
−8.983880807099595894564827988742, −8.387350750541142909385821076651, −8.021722561531263474784842673790, −7.07929769239503112485683312865, −5.82186200141112334793162938333, −4.94475463190053583814501463822, −4.33428469148017894188154576759, −3.59687271935532946703182280440, −2.32269987679036373617376536180, −1.02348586012869936201924787906,
1.02348586012869936201924787906, 2.32269987679036373617376536180, 3.59687271935532946703182280440, 4.33428469148017894188154576759, 4.94475463190053583814501463822, 5.82186200141112334793162938333, 7.07929769239503112485683312865, 8.021722561531263474784842673790, 8.387350750541142909385821076651, 8.983880807099595894564827988742
