| L(s) = 1 | − 7-s − 3·9-s − 6·11-s + 2·13-s + 3·17-s − 6·19-s − 23-s + 3·29-s − 3·31-s − 37-s + 9·41-s + 8·43-s − 4·47-s − 6·49-s − 53-s + 59-s + 8·61-s + 3·63-s + 7·67-s − 5·71-s + 6·73-s + 6·77-s + 9·81-s + 11·83-s + 4·89-s − 2·91-s − 6·97-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 9-s − 1.80·11-s + 0.554·13-s + 0.727·17-s − 1.37·19-s − 0.208·23-s + 0.557·29-s − 0.538·31-s − 0.164·37-s + 1.40·41-s + 1.21·43-s − 0.583·47-s − 6/7·49-s − 0.137·53-s + 0.130·59-s + 1.02·61-s + 0.377·63-s + 0.855·67-s − 0.593·71-s + 0.702·73-s + 0.683·77-s + 81-s + 1.20·83-s + 0.423·89-s − 0.209·91-s − 0.609·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.051914083\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.051914083\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−8.131821699878571714817457957038, −7.895907093455660908259346655755, −6.81651891786475834789657050361, −5.98644200654968908267147201428, −5.52851532679300588628541511212, −4.68254936968660354804320686566, −3.65382078628700195903836662624, −2.83193822481131617809599919331, −2.16496033805434674415326045680, −0.53727497379349778489510799132,
0.53727497379349778489510799132, 2.16496033805434674415326045680, 2.83193822481131617809599919331, 3.65382078628700195903836662624, 4.68254936968660354804320686566, 5.52851532679300588628541511212, 5.98644200654968908267147201428, 6.81651891786475834789657050361, 7.895907093455660908259346655755, 8.131821699878571714817457957038
