| L(s) = 1 | + 4·2-s + 8·4-s − 8·7-s + 8·8-s − 8·11-s − 24·13-s − 32·14-s − 4·16-s + 68·17-s − 32·22-s − 16·23-s − 96·26-s − 64·28-s + 4·31-s − 32·32-s + 272·34-s + 36·37-s − 80·41-s + 36·43-s − 64·44-s − 64·46-s − 48·47-s + 32·49-s − 192·52-s + 28·53-s − 64·56-s − 12·61-s + ⋯ |
| L(s) = 1 | + 2·2-s + 2·4-s − 8/7·7-s + 8-s − 0.727·11-s − 1.84·13-s − 2.28·14-s − 1/4·16-s + 4·17-s − 1.45·22-s − 0.695·23-s − 3.69·26-s − 2.28·28-s + 4/31·31-s − 32-s + 8·34-s + 0.972·37-s − 1.95·41-s + 0.837·43-s − 1.45·44-s − 1.39·46-s − 1.02·47-s + 0.653·49-s − 3.69·52-s + 0.528·53-s − 8/7·56-s − 0.196·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.01321458255\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01321458255\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 360 T^{3} + 4034 T^{4} + 360 p^{2} T^{5} + 32 p^{4} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 4 T + 96 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 24 T + 288 T^{2} + 5496 T^{3} + 101282 T^{4} + 5496 p^{2} T^{5} + 288 p^{4} T^{6} + 24 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 4 p T + 8 p^{2} T^{2} - 3456 p T^{3} + 1162367 T^{4} - 3456 p^{3} T^{5} + 8 p^{6} T^{6} - 4 p^{7} T^{7} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 854 T^{2} + 436035 T^{4} - 854 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 16 T + 128 T^{2} + 5088 T^{3} + 157727 T^{4} + 5088 p^{2} T^{5} + 128 p^{4} T^{6} + 16 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 1820 T^{2} + 2358 p^{2} T^{4} - 1820 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 2 T + 1197 T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 36 T + 648 T^{2} + 7092 T^{3} - 2374738 T^{4} + 7092 p^{2} T^{5} + 648 p^{4} T^{6} - 36 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 40 T + 1596 T^{2} + 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 36 T + 648 T^{2} + 38196 T^{3} - 6216658 T^{4} + 38196 p^{2} T^{5} + 648 p^{4} T^{6} - 36 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 48 T + 1152 T^{2} + 73200 T^{3} + 4183394 T^{4} + 73200 p^{2} T^{5} + 1152 p^{4} T^{6} + 48 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 28 T + 392 T^{2} + 10080 T^{3} - 9783361 T^{4} + 10080 p^{2} T^{5} + 392 p^{4} T^{6} - 28 p^{6} T^{7} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 7724 T^{2} + 30160710 T^{4} - 7724 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 6 T - 4165 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 196 T + 19208 T^{2} - 1291836 T^{3} + 82464158 T^{4} - 1291836 p^{2} T^{5} + 19208 p^{4} T^{6} - 196 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 100 T + 12432 T^{2} - 100 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 220 T + 24200 T^{2} + 2500740 T^{3} + 221959934 T^{4} + 2500740 p^{2} T^{5} + 24200 p^{4} T^{6} + 220 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 13558 T^{2} + 96326019 T^{4} - 13558 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 72 T + 2592 T^{2} + 179568 T^{3} - 6875953 T^{4} + 179568 p^{2} T^{5} + 2592 p^{4} T^{6} + 72 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 13784 T^{2} + 156530130 T^{4} - 13784 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 336 T + 56448 T^{2} + 7886928 T^{3} + 904167362 T^{4} + 7886928 p^{2} T^{5} + 56448 p^{4} T^{6} + 336 p^{6} T^{7} + p^{8} T^{8} \) |
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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
−6.60248590581133555003024030009, −6.37205541485553705775505034014, −6.16937438162635772411403036327, −5.88694490514614408789617108364, −5.62101889296480494399643917179, −5.47319964786366257792298654245, −5.23289638924613959914515970138, −5.10234296684449954292492628397, −5.08286183414683926565305931654, −4.83709309556890360648465464089, −4.26679487239581201590506874260, −4.07484484868874722973353491689, −3.95458076941939935495920308315, −3.76806879988518227732718503304, −3.35085515703443923357195076225, −3.15713446793254364413871544039, −3.03189940202765640259955302296, −2.81144472252832989845348268418, −2.57730586235707098763668822264, −2.14971236601059859212100813796, −1.95878163944609446450708006560, −1.45610180067829050382449179637, −0.975080330022372198305995870522, −0.77076811422974630780626366151, −0.01043693350369523017248407750,
0.01043693350369523017248407750, 0.77076811422974630780626366151, 0.975080330022372198305995870522, 1.45610180067829050382449179637, 1.95878163944609446450708006560, 2.14971236601059859212100813796, 2.57730586235707098763668822264, 2.81144472252832989845348268418, 3.03189940202765640259955302296, 3.15713446793254364413871544039, 3.35085515703443923357195076225, 3.76806879988518227732718503304, 3.95458076941939935495920308315, 4.07484484868874722973353491689, 4.26679487239581201590506874260, 4.83709309556890360648465464089, 5.08286183414683926565305931654, 5.10234296684449954292492628397, 5.23289638924613959914515970138, 5.47319964786366257792298654245, 5.62101889296480494399643917179, 5.88694490514614408789617108364, 6.16937438162635772411403036327, 6.37205541485553705775505034014, 6.60248590581133555003024030009
