L(s) = 1 | + 3-s − 7-s + 9-s + 11-s − 2·13-s + 6·17-s − 4·19-s − 21-s + 8·23-s + 27-s − 6·29-s − 8·31-s + 33-s + 6·37-s − 2·39-s − 6·41-s + 4·43-s − 8·47-s + 49-s + 6·51-s − 10·53-s − 4·57-s − 12·59-s + 10·61-s − 63-s + 12·67-s + 8·69-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.301·11-s − 0.554·13-s + 1.45·17-s − 0.917·19-s − 0.218·21-s + 1.66·23-s + 0.192·27-s − 1.11·29-s − 1.43·31-s + 0.174·33-s + 0.986·37-s − 0.320·39-s − 0.937·41-s + 0.609·43-s − 1.16·47-s + 1/7·49-s + 0.840·51-s − 1.37·53-s − 0.529·57-s − 1.56·59-s + 1.28·61-s − 0.125·63-s + 1.46·67-s + 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.993434079\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.993434079\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−12.62072134843622, −12.22962380179754, −11.50801815362473, −11.09745702452866, −10.80392195589868, −10.05752108002196, −9.723941619453783, −9.373974404756433, −8.931150143594728, −8.424999295748899, −7.899831654375842, −7.361278261793114, −7.215661688256847, −6.410778950495960, −6.165292076429043, −5.370937735917938, −5.033319072077900, −4.516243935400692, −3.744628860608098, −3.395458705290793, −3.067636606043924, −2.226357731044143, −1.862775799079023, −1.110673127553509, −0.4507994496043278,
0.4507994496043278, 1.110673127553509, 1.862775799079023, 2.226357731044143, 3.067636606043924, 3.395458705290793, 3.744628860608098, 4.516243935400692, 5.033319072077900, 5.370937735917938, 6.165292076429043, 6.410778950495960, 7.215661688256847, 7.361278261793114, 7.899831654375842, 8.424999295748899, 8.931150143594728, 9.373974404756433, 9.723941619453783, 10.05752108002196, 10.80392195589868, 11.09745702452866, 11.50801815362473, 12.22962380179754, 12.62072134843622