| L(s) = 1 | − 2·3-s + 4·7-s + 9-s − 6·11-s + 2·13-s + 17-s + 4·19-s − 8·21-s + 4·27-s − 4·31-s + 12·33-s − 4·37-s − 4·39-s + 6·41-s + 8·43-s + 9·49-s − 2·51-s − 6·53-s − 8·57-s + 4·61-s + 4·63-s + 8·67-s − 2·73-s − 24·77-s + 8·79-s − 11·81-s − 6·89-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.51·7-s + 1/3·9-s − 1.80·11-s + 0.554·13-s + 0.242·17-s + 0.917·19-s − 1.74·21-s + 0.769·27-s − 0.718·31-s + 2.08·33-s − 0.657·37-s − 0.640·39-s + 0.937·41-s + 1.21·43-s + 9/7·49-s − 0.280·51-s − 0.824·53-s − 1.05·57-s + 0.512·61-s + 0.503·63-s + 0.977·67-s − 0.234·73-s − 2.73·77-s + 0.900·79-s − 1.22·81-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.421653570\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.421653570\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−15.36672924692005, −14.77434425034852, −13.99834635066318, −13.89214489840302, −12.93270526742218, −12.55353300033688, −11.97207921920784, −11.26850598866836, −11.04097667132281, −10.68448108968749, −10.04072546969267, −9.312954974002456, −8.453950517255844, −8.100031990709498, −7.491070150159299, −7.035739237199059, −5.943570281995381, −5.667205781469574, −5.052646513213127, −4.789859151280367, −3.873934299723278, −2.942029838858224, −2.202686268899869, −1.315874239607975, −0.5375578424519806,
0.5375578424519806, 1.315874239607975, 2.202686268899869, 2.942029838858224, 3.873934299723278, 4.789859151280367, 5.052646513213127, 5.667205781469574, 5.943570281995381, 7.035739237199059, 7.491070150159299, 8.100031990709498, 8.453950517255844, 9.312954974002456, 10.04072546969267, 10.68448108968749, 11.04097667132281, 11.26850598866836, 11.97207921920784, 12.55353300033688, 12.93270526742218, 13.89214489840302, 13.99834635066318, 14.77434425034852, 15.36672924692005
