| L(s) = 1 | + 640·11-s − 1.40e3·13-s − 1.45e4·23-s − 1.51e5·25-s − 1.31e4·37-s − 7.09e5·47-s − 1.64e6·49-s − 3.06e6·59-s − 2.64e5·61-s − 7.98e6·71-s + 1.63e6·73-s − 1.84e7·83-s + 6.36e6·97-s − 6.74e7·107-s − 1.19e7·109-s − 2.56e7·121-s + ⋯ |
| L(s) = 1 | + 0.144·11-s − 0.176·13-s − 0.250·23-s − 1.94·25-s − 0.0427·37-s − 0.997·47-s − 2.00·49-s − 1.94·59-s − 0.149·61-s − 2.64·71-s + 0.491·73-s − 3.53·83-s + 0.708·97-s − 5.32·107-s − 0.886·109-s − 1.31·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_2^2 \wr C_2$ | \( 1 + 151572 T^{2} + 568112294 p^{2} T^{4} + 151572 p^{14} T^{6} + p^{28} T^{8} \) |
| 7 | $C_2^2 \wr C_2$ | \( 1 + 1648988 T^{2} + 1526127215334 T^{4} + 1648988 p^{14} T^{6} + p^{28} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 320 T + 12973958 T^{2} - 320 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 13 | $D_{4}$ | \( ( 1 + 700 T + 1655046 p T^{2} + 700 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 17 | $C_2^2 \wr C_2$ | \( 1 - 53528764 T^{2} + 94005192820378182 T^{4} - 53528764 p^{14} T^{6} + p^{28} T^{8} \) |
| 19 | $C_2^2 \wr C_2$ | \( 1 + 627281900 T^{2} - 415796730785898858 T^{4} + 627281900 p^{14} T^{6} + p^{28} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 7296 T + 4220360398 T^{2} + 7296 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 + 53499213492 T^{2} + \)\(12\!\cdots\!78\)\( T^{4} + 53499213492 p^{14} T^{6} + p^{28} T^{8} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 - 25861402372 T^{2} + \)\(16\!\cdots\!38\)\( T^{4} - 25861402372 p^{14} T^{6} + p^{28} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 6580 T + 189770474430 T^{2} + 6580 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 + 210306458340 T^{2} + \)\(53\!\cdots\!22\)\( T^{4} + 210306458340 p^{14} T^{6} + p^{28} T^{8} \) |
| 43 | $C_2^2 \wr C_2$ | \( 1 + 564493630604 T^{2} + \)\(22\!\cdots\!02\)\( T^{4} + 564493630604 p^{14} T^{6} + p^{28} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 7552 p T + 1021319166110 T^{2} + 7552 p^{8} T^{3} + p^{14} T^{4} )^{2} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 + 1048237870932 T^{2} + \)\(39\!\cdots\!94\)\( T^{4} + 1048237870932 p^{14} T^{6} + p^{28} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 1531520 T + 3126722309414 T^{2} + 1531520 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 132132 T + 5537699450798 T^{2} + 132132 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 + 11408315918636 T^{2} + \)\(78\!\cdots\!82\)\( T^{4} + 11408315918636 p^{14} T^{6} + p^{28} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 3991040 T + 19372360924526 T^{2} + 3991040 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 816300 T + 12256995211094 T^{2} - 816300 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 79 | $C_2^2 \wr C_2$ | \( 1 + 38820993405500 T^{2} + \)\(93\!\cdots\!62\)\( T^{4} + 38820993405500 p^{14} T^{6} + p^{28} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 9213504 T + 51757917911158 T^{2} + 9213504 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 + 145151685502116 T^{2} + \)\(91\!\cdots\!46\)\( T^{4} + 145151685502116 p^{14} T^{6} + p^{28} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 3184220 T + 130070241012102 T^{2} - 3184220 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
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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
−7.87699604522038240378340103491, −7.46399994743190994785219208463, −7.38598143876135127498856575914, −7.12981327616309301772753996581, −6.91699984105496662256038493873, −6.25222482555836593402461745840, −6.24440349726094415456251129293, −6.10431462996019185124349745175, −6.01216074452011535608526544624, −5.36757410863045912199335239610, −5.09980936186670283397054245639, −4.90253590402580990186659785304, −4.88465025325070855741720330794, −4.11623749946107116698977494871, −4.05389199245967058909526720328, −3.85718676108606708852095174272, −3.62571809780160515033681395374, −2.95633296423096475883790555856, −2.88424082003575101408689147154, −2.59813702151961986211581789452, −2.29343964576700947950227868995, −1.72417255694700988236534701909, −1.41004250329267564024797449935, −1.33122603429326206830311582177, −1.16780953143460500740103169262, 0, 0, 0, 0,
1.16780953143460500740103169262, 1.33122603429326206830311582177, 1.41004250329267564024797449935, 1.72417255694700988236534701909, 2.29343964576700947950227868995, 2.59813702151961986211581789452, 2.88424082003575101408689147154, 2.95633296423096475883790555856, 3.62571809780160515033681395374, 3.85718676108606708852095174272, 4.05389199245967058909526720328, 4.11623749946107116698977494871, 4.88465025325070855741720330794, 4.90253590402580990186659785304, 5.09980936186670283397054245639, 5.36757410863045912199335239610, 6.01216074452011535608526544624, 6.10431462996019185124349745175, 6.24440349726094415456251129293, 6.25222482555836593402461745840, 6.91699984105496662256038493873, 7.12981327616309301772753996581, 7.38598143876135127498856575914, 7.46399994743190994785219208463, 7.87699604522038240378340103491
