| L(s) = 1 | + 2-s − 4-s − 3·5-s − 7-s − 3·8-s − 3·10-s + 3·13-s − 14-s − 16-s + 17-s + 6·19-s + 3·20-s − 6·23-s + 4·25-s + 3·26-s + 28-s + 29-s + 5·31-s + 5·32-s + 34-s + 3·35-s − 12·37-s + 6·38-s + 9·40-s + 2·41-s + 43-s − 6·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.34·5-s − 0.377·7-s − 1.06·8-s − 0.948·10-s + 0.832·13-s − 0.267·14-s − 1/4·16-s + 0.242·17-s + 1.37·19-s + 0.670·20-s − 1.25·23-s + 4/5·25-s + 0.588·26-s + 0.188·28-s + 0.185·29-s + 0.898·31-s + 0.883·32-s + 0.171·34-s + 0.507·35-s − 1.97·37-s + 0.973·38-s + 1.42·40-s + 0.312·41-s + 0.152·43-s − 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.904333576\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.904333576\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−12.92606092903568, −12.29253811942827, −12.23938635834460, −11.75268584113811, −11.26168357887548, −10.81154665131581, −9.980708569634507, −9.856056692645426, −9.132366449078351, −8.661987081017386, −8.168427915057969, −7.887643654720180, −7.243907532897524, −6.701138209664372, −6.171204906616953, −5.633486424327431, −5.143242863724325, −4.617829549406465, −4.039538946402450, −3.555102456116167, −3.420249151469354, −2.738704657744038, −1.854679364647363, −0.8907897708873397, −0.4461199375392742,
0.4461199375392742, 0.8907897708873397, 1.854679364647363, 2.738704657744038, 3.420249151469354, 3.555102456116167, 4.039538946402450, 4.617829549406465, 5.143242863724325, 5.633486424327431, 6.171204906616953, 6.701138209664372, 7.243907532897524, 7.887643654720180, 8.168427915057969, 8.661987081017386, 9.132366449078351, 9.856056692645426, 9.980708569634507, 10.81154665131581, 11.26168357887548, 11.75268584113811, 12.23938635834460, 12.29253811942827, 12.92606092903568
