L(s) = 1 | − 10·3-s + 18·4-s − 150·7-s + 19·9-s − 180·12-s + 110·13-s + 68·16-s − 694·19-s + 1.50e3·21-s + 620·27-s − 2.70e3·28-s − 6·31-s + 342·36-s − 4.46e3·37-s − 1.10e3·39-s − 2.95e3·43-s − 680·48-s + 1.20e4·49-s + 1.98e3·52-s + 6.94e3·57-s + 734·61-s − 2.85e3·63-s − 3.38e3·64-s − 4.47e3·67-s + 1.39e4·73-s − 1.24e4·76-s + 9.03e3·79-s + ⋯ |
L(s) = 1 | − 1.11·3-s + 9/8·4-s − 3.06·7-s + 0.234·9-s − 5/4·12-s + 0.650·13-s + 0.265·16-s − 1.92·19-s + 3.40·21-s + 0.850·27-s − 3.44·28-s − 0.00624·31-s + 0.263·36-s − 3.25·37-s − 0.723·39-s − 1.59·43-s − 0.295·48-s + 5.02·49-s + 0.732·52-s + 2.13·57-s + 0.197·61-s − 0.718·63-s − 0.826·64-s − 0.995·67-s + 2.61·73-s − 2.16·76-s + 1.44·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.3251747785\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3251747785\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | $C_2$ | \( 1 + 10 T + p^{4} T^{2} \) |
| 5 | | \( 1 \) |
good | 2 | $C_2^2$ | \( 1 - 9 p T^{2} + p^{8} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 75 T + p^{4} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 27882 T^{2} + p^{8} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 55 T + p^{4} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 84342 T^{2} + p^{8} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 347 T + p^{4} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 135818 T^{2} + p^{8} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 673962 T^{2} + p^{8} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 3 T + p^{4} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2230 T + p^{4} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 778122 T^{2} + p^{8} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 1475 T + p^{4} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 6315138 T^{2} + p^{8} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 15482538 T^{2} + p^{8} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 16148322 T^{2} + p^{8} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 367 T + p^{4} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 2235 T + p^{4} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 50586762 T^{2} + p^{8} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6970 T + p^{4} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4518 T + p^{4} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 94817858 T^{2} + p^{8} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 60166082 T^{2} + p^{8} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 4535 T + p^{4} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.93320053043451369821846427837, −13.63787329863982183911772666622, −12.71444910413686537396472802178, −12.24670424924343081835204283642, −12.23077006619931788107679070760, −11.14569491001750393313010402913, −10.83433226615739614266427360774, −10.20297185979772001219747844934, −9.911746794208443932010240305037, −8.942210756351437794241868987255, −8.527419689682406744321453917805, −7.11344847647899500042145943840, −6.75198552205730016386270607384, −6.26375134328120116119979144219, −6.13279780986547954202361990628, −5.06352790818194116330058636110, −3.67011260935714900727613187365, −3.21035613903910152399906075516, −2.10711566850081869116974467627, −0.29058888730628904624096156182,
0.29058888730628904624096156182, 2.10711566850081869116974467627, 3.21035613903910152399906075516, 3.67011260935714900727613187365, 5.06352790818194116330058636110, 6.13279780986547954202361990628, 6.26375134328120116119979144219, 6.75198552205730016386270607384, 7.11344847647899500042145943840, 8.527419689682406744321453917805, 8.942210756351437794241868987255, 9.911746794208443932010240305037, 10.20297185979772001219747844934, 10.83433226615739614266427360774, 11.14569491001750393313010402913, 12.23077006619931788107679070760, 12.24670424924343081835204283642, 12.71444910413686537396472802178, 13.63787329863982183911772666622, 14.93320053043451369821846427837