L(s) = 1 | − 2.51·2-s + 4.33·4-s − 2.87·5-s − 3.18·7-s − 5.87·8-s + 7.23·10-s − 1.31·11-s + 6.51·13-s + 8.01·14-s + 6.11·16-s − 0.932·17-s + 0.348·19-s − 12.4·20-s + 3.31·22-s − 23-s + 3.27·25-s − 16.3·26-s − 13.8·28-s + 29-s + 4.04·31-s − 3.63·32-s + 2.34·34-s + 9.16·35-s − 10.0·37-s − 0.876·38-s + 16.8·40-s + 2.12·41-s + ⋯ |
L(s) = 1 | − 1.77·2-s + 2.16·4-s − 1.28·5-s − 1.20·7-s − 2.07·8-s + 2.28·10-s − 0.397·11-s + 1.80·13-s + 2.14·14-s + 1.52·16-s − 0.226·17-s + 0.0799·19-s − 2.78·20-s + 0.706·22-s − 0.208·23-s + 0.654·25-s − 3.21·26-s − 2.60·28-s + 0.185·29-s + 0.725·31-s − 0.642·32-s + 0.402·34-s + 1.54·35-s − 1.64·37-s − 0.142·38-s + 2.67·40-s + 0.331·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
good | 2 | \( 1 + 2.51T + 2T^{2} \) |
| 5 | \( 1 + 2.87T + 5T^{2} \) |
| 7 | \( 1 + 3.18T + 7T^{2} \) |
| 11 | \( 1 + 1.31T + 11T^{2} \) |
| 13 | \( 1 - 6.51T + 13T^{2} \) |
| 17 | \( 1 + 0.932T + 17T^{2} \) |
| 19 | \( 1 - 0.348T + 19T^{2} \) |
| 31 | \( 1 - 4.04T + 31T^{2} \) |
| 37 | \( 1 + 10.0T + 37T^{2} \) |
| 41 | \( 1 - 2.12T + 41T^{2} \) |
| 43 | \( 1 + 3.91T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 + 6.92T + 53T^{2} \) |
| 59 | \( 1 - 7.17T + 59T^{2} \) |
| 61 | \( 1 + 12.1T + 61T^{2} \) |
| 67 | \( 1 - 3.95T + 67T^{2} \) |
| 71 | \( 1 - 15.7T + 71T^{2} \) |
| 73 | \( 1 - 11.0T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 - 4.56T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 - 16.5T + 97T^{2} \) |
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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
−7.940548173769178089674611456047, −7.27282652953666375883870019577, −6.45639352194466232870612858894, −6.20542833502161631817342888661, −4.80186118521589721777807419426, −3.46193153335495038191434308683, −3.37719774990725307941332522789, −2.01690776418211942705240654958, −0.862550857756657226320815830546, 0,
0.862550857756657226320815830546, 2.01690776418211942705240654958, 3.37719774990725307941332522789, 3.46193153335495038191434308683, 4.80186118521589721777807419426, 6.20542833502161631817342888661, 6.45639352194466232870612858894, 7.27282652953666375883870019577, 7.940548173769178089674611456047