| L(s) = 1 | − 7-s − 11-s + 4·13-s + 3·17-s + 19-s + 9·23-s − 6·29-s + 10·31-s + 37-s − 9·41-s − 4·43-s + 3·47-s − 6·49-s − 12·53-s − 3·59-s − 4·61-s − 4·67-s + 3·71-s + 4·73-s + 77-s − 11·79-s − 6·83-s − 4·91-s + 97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 0.301·11-s + 1.10·13-s + 0.727·17-s + 0.229·19-s + 1.87·23-s − 1.11·29-s + 1.79·31-s + 0.164·37-s − 1.40·41-s − 0.609·43-s + 0.437·47-s − 6/7·49-s − 1.64·53-s − 0.390·59-s − 0.512·61-s − 0.488·67-s + 0.356·71-s + 0.468·73-s + 0.113·77-s − 1.23·79-s − 0.658·83-s − 0.419·91-s + 0.101·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−15.06488150608966, −14.62191069223311, −13.91187372556678, −13.37641727452974, −13.14894157469622, −12.50486492907566, −11.90369840632686, −11.38656744087389, −10.82163408355964, −10.44152511589446, −9.588475416650383, −9.428970546235167, −8.562743310116463, −8.196583746592956, −7.582532924172998, −6.827401955087705, −6.479557119914817, −5.759706429033250, −5.195118043615425, −4.619781842627897, −3.806638677721636, −3.115912847152286, −2.847470598816713, −1.593576420706435, −1.112862797115245, 0,
1.112862797115245, 1.593576420706435, 2.847470598816713, 3.115912847152286, 3.806638677721636, 4.619781842627897, 5.195118043615425, 5.759706429033250, 6.479557119914817, 6.827401955087705, 7.582532924172998, 8.196583746592956, 8.562743310116463, 9.428970546235167, 9.588475416650383, 10.44152511589446, 10.82163408355964, 11.38656744087389, 11.90369840632686, 12.50486492907566, 13.14894157469622, 13.37641727452974, 13.91187372556678, 14.62191069223311, 15.06488150608966
