L(s) = 1 | − 2·4-s + 7-s + 13-s + 4·16-s + 17-s − 7·19-s + 8·23-s − 2·28-s − 6·29-s + 9·37-s + 41-s + 8·43-s + 49-s − 2·52-s − 53-s + 14·59-s − 5·61-s − 8·64-s + 5·67-s − 2·68-s − 9·71-s + 7·73-s + 14·76-s − 10·79-s + 8·83-s + 8·89-s + 91-s + ⋯ |
L(s) = 1 | − 4-s + 0.377·7-s + 0.277·13-s + 16-s + 0.242·17-s − 1.60·19-s + 1.66·23-s − 0.377·28-s − 1.11·29-s + 1.47·37-s + 0.156·41-s + 1.21·43-s + 1/7·49-s − 0.277·52-s − 0.137·53-s + 1.82·59-s − 0.640·61-s − 64-s + 0.610·67-s − 0.242·68-s − 1.06·71-s + 0.819·73-s + 1.60·76-s − 1.12·79-s + 0.878·83-s + 0.847·89-s + 0.104·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−13.38109421977632, −12.86887662896832, −12.61670065638523, −12.03813741197178, −11.31559780914264, −10.92385283935262, −10.69150071689796, −9.887292532228544, −9.575352214064379, −8.938300875636275, −8.719027857302978, −8.193613859575464, −7.635746950892076, −7.223667349326651, −6.513135365361059, −5.975258758675756, −5.514633282791021, −4.938374567867676, −4.476247941789253, −3.980333447056012, −3.540838841993761, −2.722642447079656, −2.221458405155250, −1.317893848768237, −0.8283304673150226, 0,
0.8283304673150226, 1.317893848768237, 2.221458405155250, 2.722642447079656, 3.540838841993761, 3.980333447056012, 4.476247941789253, 4.938374567867676, 5.514633282791021, 5.975258758675756, 6.513135365361059, 7.223667349326651, 7.635746950892076, 8.193613859575464, 8.719027857302978, 8.938300875636275, 9.575352214064379, 9.887292532228544, 10.69150071689796, 10.92385283935262, 11.31559780914264, 12.03813741197178, 12.61670065638523, 12.86887662896832, 13.38109421977632