| L(s) = 1 | + 7-s − 3·9-s − 4·13-s + 4·17-s + 2·19-s + 23-s + 6·29-s + 2·31-s − 6·37-s − 2·41-s + 4·43-s − 6·47-s + 49-s + 6·53-s + 2·61-s − 3·63-s − 4·67-s + 12·71-s + 10·73-s − 8·79-s + 9·81-s − 6·83-s − 4·91-s − 8·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 9-s − 1.10·13-s + 0.970·17-s + 0.458·19-s + 0.208·23-s + 1.11·29-s + 0.359·31-s − 0.986·37-s − 0.312·41-s + 0.609·43-s − 0.875·47-s + 1/7·49-s + 0.824·53-s + 0.256·61-s − 0.377·63-s − 0.488·67-s + 1.42·71-s + 1.17·73-s − 0.900·79-s + 81-s − 0.658·83-s − 0.419·91-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−12.98831408163755, −12.47123325017928, −11.97883190118648, −11.87429660012507, −11.25987089054754, −10.74878417865510, −10.27890147164758, −9.803153114254437, −9.425045265439353, −8.812109401010473, −8.266583517936898, −8.070709823294645, −7.396783359479061, −6.989365936770908, −6.447551390762921, −5.817487626811438, −5.308106381526214, −5.038080419099951, −4.472951244457007, −3.721349126974239, −3.208700094593413, −2.675481132872702, −2.236615368140946, −1.401292836789335, −0.7786245842935377, 0,
0.7786245842935377, 1.401292836789335, 2.236615368140946, 2.675481132872702, 3.208700094593413, 3.721349126974239, 4.472951244457007, 5.038080419099951, 5.308106381526214, 5.817487626811438, 6.447551390762921, 6.989365936770908, 7.396783359479061, 8.070709823294645, 8.266583517936898, 8.812109401010473, 9.425045265439353, 9.803153114254437, 10.27890147164758, 10.74878417865510, 11.25987089054754, 11.87429660012507, 11.97883190118648, 12.47123325017928, 12.98831408163755
