| L(s) = 1 | + 3-s + 4·7-s − 2·9-s + 3·11-s − 17-s + 7·19-s + 4·21-s + 4·23-s − 5·27-s − 8·29-s + 4·31-s + 3·33-s − 4·37-s − 3·41-s + 8·43-s + 9·49-s − 51-s + 12·53-s + 7·57-s + 8·59-s + 4·61-s − 8·63-s − 9·67-s + 4·69-s + 16·71-s − 11·73-s + 12·77-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.51·7-s − 2/3·9-s + 0.904·11-s − 0.242·17-s + 1.60·19-s + 0.872·21-s + 0.834·23-s − 0.962·27-s − 1.48·29-s + 0.718·31-s + 0.522·33-s − 0.657·37-s − 0.468·41-s + 1.21·43-s + 9/7·49-s − 0.140·51-s + 1.64·53-s + 0.927·57-s + 1.04·59-s + 0.512·61-s − 1.00·63-s − 1.09·67-s + 0.481·69-s + 1.89·71-s − 1.28·73-s + 1.36·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.260298056\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.260298056\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 + 13 T + p T^{2} \) |
| 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
−8.061632372862743115667091017793, −7.43100129039840738482630785534, −6.83694345392637197451575922422, −5.59490587668282905141289002019, −5.33686413198391101661533754339, −4.32636959483255040005342843011, −3.62374274040571084531264233335, −2.72662471884265745613618932294, −1.83224079601896310337987946848, −0.973725220887259086889476267910,
0.973725220887259086889476267910, 1.83224079601896310337987946848, 2.72662471884265745613618932294, 3.62374274040571084531264233335, 4.32636959483255040005342843011, 5.33686413198391101661533754339, 5.59490587668282905141289002019, 6.83694345392637197451575922422, 7.43100129039840738482630785534, 8.061632372862743115667091017793
