| L(s) = 1 | − 8·13-s + 2·25-s + 32·37-s + 22·49-s − 16·61-s + 4·73-s − 20·97-s + 64·109-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 12·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 2.21·13-s + 2/5·25-s + 5.26·37-s + 22/7·49-s − 2.04·61-s + 0.468·73-s − 2.03·97-s + 6.13·109-s + 0.909·121-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 − 0.923·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.994309508\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.994309508\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 35 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 146 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 139 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) |
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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
−7.87317401467508419614123921028, −7.68698888432335792429726619968, −7.56588713504471124368803883400, −7.44301884967625988068297462613, −7.30014750547882064027662363153, −6.84056262076942450768805486135, −6.66182779471982331097801526214, −6.24481860918258213460716285256, −5.90713250724622261853421486753, −5.84604293199146255569440679473, −5.84463299728445885488654169889, −5.02583895672560489429800113309, −5.02174898797827268788226781339, −4.77000461558026142111658769539, −4.40479283008061550561707377289, −4.12048460063782532521739283803, −4.11412279329988354133145093741, −3.44365976333848894796798811627, −3.10651413491457010203545955531, −2.68817947782363692601541890126, −2.44175318077765701730775770466, −2.39821098636516642662766256821, −1.76990784687708791770356965656, −1.00369578083096626536781980315, −0.61251263848817942301661912056,
0.61251263848817942301661912056, 1.00369578083096626536781980315, 1.76990784687708791770356965656, 2.39821098636516642662766256821, 2.44175318077765701730775770466, 2.68817947782363692601541890126, 3.10651413491457010203545955531, 3.44365976333848894796798811627, 4.11412279329988354133145093741, 4.12048460063782532521739283803, 4.40479283008061550561707377289, 4.77000461558026142111658769539, 5.02174898797827268788226781339, 5.02583895672560489429800113309, 5.84463299728445885488654169889, 5.84604293199146255569440679473, 5.90713250724622261853421486753, 6.24481860918258213460716285256, 6.66182779471982331097801526214, 6.84056262076942450768805486135, 7.30014750547882064027662363153, 7.44301884967625988068297462613, 7.56588713504471124368803883400, 7.68698888432335792429726619968, 7.87317401467508419614123921028
