L(s) = 1 | − 3-s − 7-s + 9-s − 11-s + 2·13-s + 2·17-s + 21-s − 27-s − 2·29-s + 4·31-s + 33-s − 6·37-s − 2·39-s + 6·41-s + 12·43-s + 4·47-s + 49-s − 2·51-s − 6·53-s + 4·59-s − 10·61-s − 63-s − 4·67-s − 8·71-s + 2·73-s + 77-s + 81-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s + 0.485·17-s + 0.218·21-s − 0.192·27-s − 0.371·29-s + 0.718·31-s + 0.174·33-s − 0.986·37-s − 0.320·39-s + 0.937·41-s + 1.82·43-s + 0.583·47-s + 1/7·49-s − 0.280·51-s − 0.824·53-s + 0.520·59-s − 1.28·61-s − 0.125·63-s − 0.488·67-s − 0.949·71-s + 0.234·73-s + 0.113·77-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−14.12967287907512, −13.56039568475770, −12.97855067611371, −12.63465523694006, −12.13291644158036, −11.67220369917871, −11.02458311177528, −10.71787417365258, −10.18398579223322, −9.692725497259872, −9.108372354085203, −8.662658684044453, −7.995063905985202, −7.394180571083402, −7.107906313130052, −6.234051791391072, −5.952525316523978, −5.503152844490989, −4.730938516591213, −4.273422535963986, −3.609642311280191, −2.998258997288927, −2.355877027887232, −1.486510614646486, −0.8545146071009099, 0,
0.8545146071009099, 1.486510614646486, 2.355877027887232, 2.998258997288927, 3.609642311280191, 4.273422535963986, 4.730938516591213, 5.503152844490989, 5.952525316523978, 6.234051791391072, 7.107906313130052, 7.394180571083402, 7.995063905985202, 8.662658684044453, 9.108372354085203, 9.692725497259872, 10.18398579223322, 10.71787417365258, 11.02458311177528, 11.67220369917871, 12.13291644158036, 12.63465523694006, 12.97855067611371, 13.56039568475770, 14.12967287907512