| L(s) = 1 | − 2·3-s + 4·7-s + 9-s − 6·11-s − 2·13-s − 17-s − 8·21-s − 5·25-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 − 4·61-s + 4·63-s + 8·67-s + 2·73-s + 10·75-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 − 1.74·21-s − 25-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 − 0.512·61-s + 0.503·63-s + 0.977·67-s + 0.234·73-s + 1.15·75-s − 2.73·77-s + 0.900·79-s − 1.22·81-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 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
−13.94882752608187, −13.50081490237200, −13.00511972569029, −12.43389573915547, −11.92819103584441, −11.55277315820549, −11.01789038061894, −10.82041020867042, −10.14405514945507, −9.879624951384908, −8.969776711080503, −8.375841002392209, −7.951442582408140, −7.583055859909609, −6.973888797965078, −6.329911031894506, −5.575062289151808, −5.358371790250782, −4.895206529346779, −4.506825894912392, −3.673207104422584, −2.809317228898061, −2.174746265832833, −1.669715524639437, −0.6662180627591938, 0,
0.6662180627591938, 1.669715524639437, 2.174746265832833, 2.809317228898061, 3.673207104422584, 4.506825894912392, 4.895206529346779, 5.358371790250782, 5.575062289151808, 6.329911031894506, 6.973888797965078, 7.583055859909609, 7.951442582408140, 8.375841002392209, 8.969776711080503, 9.879624951384908, 10.14405514945507, 10.82041020867042, 11.01789038061894, 11.55277315820549, 11.92819103584441, 12.43389573915547, 13.00511972569029, 13.50081490237200, 13.94882752608187
