| L(s) = 1 | − 2·7-s + 48·19-s − 25·25-s + 105·31-s + 47·37-s + 122·43-s − 45·49-s + 195·61-s + 109·67-s + 240·73-s − 131·79-s − 351·103-s + 71·109-s + 121·121-s + 127-s + 131-s − 96·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 337·169-s + 173-s + 50·175-s + ⋯ |
| L(s) = 1 | − 2/7·7-s + 2.52·19-s − 25-s + 3.38·31-s + 1.27·37-s + 2.83·43-s − 0.918·49-s + 3.19·61-s + 1.62·67-s + 3.28·73-s − 1.65·79-s − 3.40·103-s + 0.651·109-s + 121-s + 0.00787·127-s + 0.00763·131-s − 0.721·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 1.99·169-s + 0.00578·173-s + 2/7·175-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 571536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571536 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.369766772\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.369766772\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 2 T + p^{2} T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - p^{2} T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + p^{2} T^{2} )( 1 + T + p^{2} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 37 T + p^{2} T^{2} )( 1 - 11 T + p^{2} T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - p^{2} T^{2} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 59 T + p^{2} T^{2} )( 1 - 46 T + p^{2} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 73 T + p^{2} T^{2} )( 1 + 26 T + p^{2} T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 61 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - p^{2} T^{2} + p^{4} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 121 T + p^{2} T^{2} )( 1 - 74 T + p^{2} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 122 T + p^{2} T^{2} )( 1 + 13 T + p^{2} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 143 T + p^{2} T^{2} )( 1 - 97 T + p^{2} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 11 T + p^{2} T^{2} )( 1 + 142 T + p^{2} T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 169 T + p^{2} T^{2} )( 1 + 169 T + p^{2} T^{2} ) \) |
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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
−10.02095862710318444675731854851, −9.901596327104109752522462137889, −9.572419445298507969830384001570, −9.382951432796655410231656925479, −8.380467038711262573335464690038, −8.334178410316347508708379663016, −7.73729636261113171769364985928, −7.46542230588668859006428427653, −6.73697380326256297213565021866, −6.55411938175752420868010541701, −5.81106448956747275137085565643, −5.55378113429104500795746660793, −4.99234295592989487063688917126, −4.40636601513099559497744753848, −3.90144159699070434728390289921, −3.33796403137262253926594716758, −2.63588596354056178943296316536, −2.35703640745463054226798789130, −0.958653013732935254240511749481, −0.887739736429597782747576954097,
0.887739736429597782747576954097, 0.958653013732935254240511749481, 2.35703640745463054226798789130, 2.63588596354056178943296316536, 3.33796403137262253926594716758, 3.90144159699070434728390289921, 4.40636601513099559497744753848, 4.99234295592989487063688917126, 5.55378113429104500795746660793, 5.81106448956747275137085565643, 6.55411938175752420868010541701, 6.73697380326256297213565021866, 7.46542230588668859006428427653, 7.73729636261113171769364985928, 8.334178410316347508708379663016, 8.380467038711262573335464690038, 9.382951432796655410231656925479, 9.572419445298507969830384001570, 9.901596327104109752522462137889, 10.02095862710318444675731854851
