| L(s) = 1 | − 8·16-s + 28·31-s − 52·61-s − 9·81-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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 + 239-s + ⋯ |
| L(s) = 1 | − 2·16-s + 5.02·31-s − 6.65·61-s − 81-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 + 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 + 0.0646·239-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7195562622\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7195562622\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2}( 1 + p T + p T^{2} )^{2} \) |
| 7 | $C_2^3$ | \( 1 + 23 T^{4} + p^{4} T^{8} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 13 | $C_2^3$ | \( 1 - 337 T^{4} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{4} \) |
| 37 | $C_2^3$ | \( 1 - 2062 T^{4} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 43 | $C_2^3$ | \( 1 + 23 T^{4} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{4} \) |
| 67 | $C_2^3$ | \( 1 + 2903 T^{4} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 - 8542 T^{4} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 97 | $C_2^3$ | \( 1 + 9743 T^{4} + p^{4} T^{8} \) |
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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
−10.91716168672346151599909082160, −10.53054949609946400449155059232, −10.15975893520387179183476691193, −9.952799119937312366033459277162, −9.772617469579423903268941548710, −9.231432884730583854010018457341, −9.036581819683474654658054851719, −8.681011624422328193694113894818, −8.550953013005702770550571426653, −7.994397841346464287665603950071, −7.77393031490225384892523888350, −7.49437130344566530573441106044, −6.98889702694688790200163282196, −6.68718685796925635333848569441, −6.30718475875464681992262523024, −6.01022959873947222877292291735, −5.96228250188154202928945484523, −4.86441124038730884874123218676, −4.70269006944242844820140091690, −4.64167692735357968724799275556, −4.17428157854659057736214700116, −3.36245573472265835272861328486, −2.70868322794637484948616079394, −2.65308386127335085338012855862, −1.51080945135387386955239285171,
1.51080945135387386955239285171, 2.65308386127335085338012855862, 2.70868322794637484948616079394, 3.36245573472265835272861328486, 4.17428157854659057736214700116, 4.64167692735357968724799275556, 4.70269006944242844820140091690, 4.86441124038730884874123218676, 5.96228250188154202928945484523, 6.01022959873947222877292291735, 6.30718475875464681992262523024, 6.68718685796925635333848569441, 6.98889702694688790200163282196, 7.49437130344566530573441106044, 7.77393031490225384892523888350, 7.994397841346464287665603950071, 8.550953013005702770550571426653, 8.681011624422328193694113894818, 9.036581819683474654658054851719, 9.231432884730583854010018457341, 9.772617469579423903268941548710, 9.952799119937312366033459277162, 10.15975893520387179183476691193, 10.53054949609946400449155059232, 10.91716168672346151599909082160
