L(s) = 1 | + 3-s − 5-s + 7-s + 9-s − 4·11-s + 2·13-s − 15-s + 2·17-s + 4·19-s + 21-s + 8·23-s + 25-s + 27-s + 2·29-s − 4·33-s − 35-s − 6·37-s + 2·39-s − 6·41-s − 4·43-s − 45-s + 49-s + 2·51-s + 10·53-s + 4·55-s + 4·57-s + 12·59-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 1.20·11-s + 0.554·13-s − 0.258·15-s + 0.485·17-s + 0.917·19-s + 0.218·21-s + 1.66·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s − 0.696·33-s − 0.169·35-s − 0.986·37-s + 0.320·39-s − 0.937·41-s − 0.609·43-s − 0.149·45-s + 1/7·49-s + 0.280·51-s + 1.37·53-s + 0.539·55-s + 0.529·57-s + 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.437554755\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.437554755\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.067792919413443572268908629009, −7.29466767342624434535662347402, −6.91165704585120918415943435579, −5.66837498117775023160756110431, −5.13429871643228048050619453893, −4.41113790939785504303303719119, −3.30832192254682250069710858165, −3.00705106888789716184359223974, −1.83674354485910906002811308802, −0.796805449064536655798640765276,
0.796805449064536655798640765276, 1.83674354485910906002811308802, 3.00705106888789716184359223974, 3.30832192254682250069710858165, 4.41113790939785504303303719119, 5.13429871643228048050619453893, 5.66837498117775023160756110431, 6.91165704585120918415943435579, 7.29466767342624434535662347402, 8.067792919413443572268908629009