L(s) = 1 | + 2-s + 4-s + 5-s + 8-s + 10-s + 4·11-s − 6·13-s + 16-s + 6·17-s + 4·19-s + 20-s + 4·22-s − 23-s + 25-s − 6·26-s + 6·29-s − 8·31-s + 32-s + 6·34-s + 6·37-s + 4·38-s + 40-s − 10·41-s + 4·43-s + 4·44-s − 46-s + 8·47-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s + 0.316·10-s + 1.20·11-s − 1.66·13-s + 1/4·16-s + 1.45·17-s + 0.917·19-s + 0.223·20-s + 0.852·22-s − 0.208·23-s + 1/5·25-s − 1.17·26-s + 1.11·29-s − 1.43·31-s + 0.176·32-s + 1.02·34-s + 0.986·37-s + 0.648·38-s + 0.158·40-s − 1.56·41-s + 0.609·43-s + 0.603·44-s − 0.147·46-s + 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.256982269\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.256982269\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 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 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−9.382786779937371819987783736909, −8.232215862797765904742560189080, −7.29356680017341983104579693045, −6.83167002982379912689199133265, −5.71323335841138265866627321199, −5.23821231340755244483380240660, −4.23790248325532820999782331121, −3.32197498177051620753450033791, −2.36866993182259818840915365989, −1.17351616216495372278974762865,
1.17351616216495372278974762865, 2.36866993182259818840915365989, 3.32197498177051620753450033791, 4.23790248325532820999782331121, 5.23821231340755244483380240660, 5.71323335841138265866627321199, 6.83167002982379912689199133265, 7.29356680017341983104579693045, 8.232215862797765904742560189080, 9.382786779937371819987783736909