L(s) = 1 | + 2-s + 4-s − 1.73·5-s + 3.46·7-s + 8-s − 1.73·10-s − 1.73·13-s + 3.46·14-s + 16-s − 3·17-s + 3.46·19-s − 1.73·20-s − 8.66·23-s − 2.00·25-s − 1.73·26-s + 3.46·28-s − 6·29-s − 8·31-s + 32-s − 3·34-s − 5.99·35-s − 2·37-s + 3.46·38-s − 1.73·40-s − 3·41-s + 10.3·43-s − 8.66·46-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.774·5-s + 1.30·7-s + 0.353·8-s − 0.547·10-s − 0.480·13-s + 0.925·14-s + 0.250·16-s − 0.727·17-s + 0.794·19-s − 0.387·20-s − 1.80·23-s − 0.400·25-s − 0.339·26-s + 0.654·28-s − 1.11·29-s − 1.43·31-s + 0.176·32-s − 0.514·34-s − 1.01·35-s − 0.328·37-s + 0.561·38-s − 0.273·40-s − 0.468·41-s + 1.58·43-s − 1.27·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 5 | \( 1 + 1.73T + 5T^{2} \) |
| 7 | \( 1 - 3.46T + 7T^{2} \) |
| 13 | \( 1 + 1.73T + 13T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 - 3.46T + 19T^{2} \) |
| 23 | \( 1 + 8.66T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 + 5.19T + 53T^{2} \) |
| 59 | \( 1 + 6.92T + 59T^{2} \) |
| 61 | \( 1 + 8.66T + 61T^{2} \) |
| 67 | \( 1 + 5T + 67T^{2} \) |
| 71 | \( 1 + 3.46T + 71T^{2} \) |
| 73 | \( 1 + 3.46T + 73T^{2} \) |
| 79 | \( 1 - 1.73T + 79T^{2} \) |
| 83 | \( 1 + 15T + 83T^{2} \) |
| 89 | \( 1 - 3.46T + 89T^{2} \) |
| 97 | \( 1 + T + 97T^{2} \) |
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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
−7.50342588026418935365678157802, −7.25961451435885021238256963431, −5.95385174751376845883640187731, −5.54305054608083847952376484988, −4.57265256431894827846677782289, −4.18586450923328473455144640197, −3.40762888043227374481608725280, −2.25129409898673973499976289883, −1.60309518495805361872190773866, 0,
1.60309518495805361872190773866, 2.25129409898673973499976289883, 3.40762888043227374481608725280, 4.18586450923328473455144640197, 4.57265256431894827846677782289, 5.54305054608083847952376484988, 5.95385174751376845883640187731, 7.25961451435885021238256963431, 7.50342588026418935365678157802