| L(s) = 1 | + 8·9-s + 24·17-s − 2·25-s − 32·41-s + 8·49-s − 24·73-s + 30·81-s − 24·89-s + 40·97-s − 40·113-s − 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 192·153-s + 157-s + 163-s + 167-s + 44·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 8/3·9-s + 5.82·17-s − 2/5·25-s − 4.99·41-s + 8/7·49-s − 2.80·73-s + 10/3·81-s − 2.54·89-s + 4.06·97-s − 3.76·113-s − 1.81·121-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 + 15.5·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.38·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.870557824\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.870557824\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 3 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 84 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 92 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 116 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 134 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
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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
−7.70348213657919961821157465824, −7.29314904572308201147759402196, −7.24182939589077284796216240559, −6.95263691217372191344450813372, −6.77848204638477211022921944776, −6.64955722366092072653421203520, −5.97626124721327393839958169505, −5.88570515159656489678736126700, −5.82232411586035505478478461304, −5.34612939172440329244279706201, −5.11113362790679925866945707521, −5.01126945023178015261143991850, −4.92164590112434797275952424343, −4.15515308599506132677908864972, −4.08198216291624469518957446726, −3.93747824915481641699339942293, −3.51306791888257889842289374559, −3.19568186379880682856062887964, −3.15043352519687779289810720227, −2.86764407793390778683761792912, −2.08244044112979950983335092636, −1.51534072924660585125836413196, −1.47020581996749346135711012764, −1.36526500724311728966008174151, −0.69360634084577674764769493279,
0.69360634084577674764769493279, 1.36526500724311728966008174151, 1.47020581996749346135711012764, 1.51534072924660585125836413196, 2.08244044112979950983335092636, 2.86764407793390778683761792912, 3.15043352519687779289810720227, 3.19568186379880682856062887964, 3.51306791888257889842289374559, 3.93747824915481641699339942293, 4.08198216291624469518957446726, 4.15515308599506132677908864972, 4.92164590112434797275952424343, 5.01126945023178015261143991850, 5.11113362790679925866945707521, 5.34612939172440329244279706201, 5.82232411586035505478478461304, 5.88570515159656489678736126700, 5.97626124721327393839958169505, 6.64955722366092072653421203520, 6.77848204638477211022921944776, 6.95263691217372191344450813372, 7.24182939589077284796216240559, 7.29314904572308201147759402196, 7.70348213657919961821157465824
