| L(s) = 1 | − 16·25-s + 28·49-s − 64·73-s + 32·97-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 20·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | − 3.19·25-s + 4·49-s − 7.49·73-s + 3.24·97-s + 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.53·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.516127581\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.516127581\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 - 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 80 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 + 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 - 160 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.25956749918183102744458763751, −6.89745690342927009606610399059, −6.51326974067145458704143866011, −6.17180035881777150864930677334, −5.93574652185219288794978444693, −5.92803941546673238328386711729, −5.84475588624455757937656780684, −5.49324493975543760523002417614, −5.42256329844691614814054598713, −4.74533844902306284267694050419, −4.68759441609376948252075447161, −4.60311563185912939554430170615, −4.07530358119123125094366381344, −4.04312779743097331899890777333, −3.87559949350476575447948535690, −3.42612492911108682584712036061, −3.25300848910838711270412377231, −2.93349183556305082939007592302, −2.47993195824009665682586058296, −2.38787260501486819090338192923, −2.03394373135791130806666996269, −1.59120727489496649931361577032, −1.47576052914384696253817129458, −0.797260479255646131653182496628, −0.30595094782506241078071687660,
0.30595094782506241078071687660, 0.797260479255646131653182496628, 1.47576052914384696253817129458, 1.59120727489496649931361577032, 2.03394373135791130806666996269, 2.38787260501486819090338192923, 2.47993195824009665682586058296, 2.93349183556305082939007592302, 3.25300848910838711270412377231, 3.42612492911108682584712036061, 3.87559949350476575447948535690, 4.04312779743097331899890777333, 4.07530358119123125094366381344, 4.60311563185912939554430170615, 4.68759441609376948252075447161, 4.74533844902306284267694050419, 5.42256329844691614814054598713, 5.49324493975543760523002417614, 5.84475588624455757937656780684, 5.92803941546673238328386711729, 5.93574652185219288794978444693, 6.17180035881777150864930677334, 6.51326974067145458704143866011, 6.89745690342927009606610399059, 7.25956749918183102744458763751
