L(s) = 1 | − 5-s + 7-s − 4·11-s − 2·13-s − 2·17-s − 4·19-s − 8·23-s + 25-s + 2·29-s − 35-s + 6·37-s + 6·41-s + 4·43-s + 49-s + 10·53-s + 4·55-s + 12·59-s + 14·61-s + 2·65-s + 12·67-s − 8·71-s + 10·73-s − 4·77-s − 16·79-s − 12·83-s + 2·85-s − 10·89-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.377·7-s − 1.20·11-s − 0.554·13-s − 0.485·17-s − 0.917·19-s − 1.66·23-s + 1/5·25-s + 0.371·29-s − 0.169·35-s + 0.986·37-s + 0.937·41-s + 0.609·43-s + 1/7·49-s + 1.37·53-s + 0.539·55-s + 1.56·59-s + 1.79·61-s + 0.248·65-s + 1.46·67-s − 0.949·71-s + 1.17·73-s − 0.455·77-s − 1.80·79-s − 1.31·83-s + 0.216·85-s − 1.05·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.184645399\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.184645399\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 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
−8.294828713181742949184508016872, −7.56874653941833282855481017561, −6.95035037711927005887175618822, −5.97281723123870342430161791428, −5.34536264668120410165998790926, −4.41149999753964532859466601797, −3.95920776380444365991400028821, −2.62078960744102897214509959409, −2.15643106585853480886830598139, −0.56480077807673492365674115593,
0.56480077807673492365674115593, 2.15643106585853480886830598139, 2.62078960744102897214509959409, 3.95920776380444365991400028821, 4.41149999753964532859466601797, 5.34536264668120410165998790926, 5.97281723123870342430161791428, 6.95035037711927005887175618822, 7.56874653941833282855481017561, 8.294828713181742949184508016872