| L(s) = 1 | + 14·3-s − 42·5-s + 894·9-s − 7.42e3·11-s + 2.36e4·13-s − 588·15-s − 1.57e4·17-s − 2.66e4·19-s − 3.26e4·23-s + 1.24e5·25-s − 2.64e4·27-s − 3.16e5·29-s + 1.80e5·31-s − 1.03e5·33-s + 4.58e4·37-s + 3.31e5·39-s − 6.43e5·41-s + 2.04e6·43-s − 3.75e4·45-s − 1.66e6·47-s − 2.21e5·51-s + 4.10e5·53-s + 3.11e5·55-s − 3.72e5·57-s − 1.70e6·59-s + 5.47e5·61-s − 9.93e5·65-s + ⋯ |
| L(s) = 1 | + 0.299·3-s − 0.150·5-s + 0.408·9-s − 1.68·11-s + 2.98·13-s − 0.0449·15-s − 0.779·17-s − 0.890·19-s − 0.559·23-s + 1.59·25-s − 0.258·27-s − 2.40·29-s + 1.08·31-s − 0.503·33-s + 0.148·37-s + 0.894·39-s − 1.45·41-s + 3.92·43-s − 0.0614·45-s − 2.34·47-s − 0.233·51-s + 0.378·53-s + 0.252·55-s − 0.266·57-s − 1.07·59-s + 0.308·61-s − 0.448·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.09191242526\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09191242526\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 14 T - 698 T^{2} + 16240 p T^{3} - 490265 p^{2} T^{4} + 16240 p^{8} T^{5} - 698 p^{14} T^{6} - 14 p^{21} T^{7} + p^{28} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 + 42 T - 123166 T^{2} - 263088 p T^{3} + 374647071 p^{2} T^{4} - 263088 p^{8} T^{5} - 123166 p^{14} T^{6} + 42 p^{21} T^{7} + p^{28} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 7428 T + 8632202 T^{2} + 56219857920 T^{3} + 711291094304619 T^{4} + 56219857920 p^{7} T^{5} + 8632202 p^{14} T^{6} + 7428 p^{21} T^{7} + p^{28} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 70 p^{2} T + 160198410 T^{2} - 70 p^{9} T^{3} + p^{14} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 15792 T - 16280606 p T^{2} - 4651056365760 T^{3} + 6130718726782755 T^{4} - 4651056365760 p^{7} T^{5} - 16280606 p^{15} T^{6} + 15792 p^{21} T^{7} + p^{28} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 26614 T - 1253942090 T^{2} + 4644239023312 T^{3} + 2418273122215699231 T^{4} + 4644239023312 p^{7} T^{5} - 1253942090 p^{14} T^{6} + 26614 p^{21} T^{7} + p^{28} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 32640 T + 2057178482 T^{2} - 254639647088640 T^{3} - 14236548990664871085 T^{4} - 254639647088640 p^{7} T^{5} + 2057178482 p^{14} T^{6} + 32640 p^{21} T^{7} + p^{28} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 158016 T + 39988772806 T^{2} + 158016 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 180740 T - 5817373358 T^{2} + 2989603578895360 T^{3} - \)\(17\!\cdots\!57\)\( T^{4} + 2989603578895360 p^{7} T^{5} - 5817373358 p^{14} T^{6} - 180740 p^{21} T^{7} + p^{28} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 45824 T + 20185374250 T^{2} + 9529068243880960 T^{3} - \)\(88\!\cdots\!21\)\( T^{4} + 9529068243880960 p^{7} T^{5} + 20185374250 p^{14} T^{6} - 45824 p^{21} T^{7} + p^{28} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 321720 T + 181439440606 T^{2} + 321720 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 1023868 T + 671194246566 T^{2} - 1023868 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 1665972 T + 1099984870706 T^{2} + 1103259291706623744 T^{3} + \)\(11\!\cdots\!23\)\( T^{4} + 1103259291706623744 p^{7} T^{5} + 1099984870706 p^{14} T^{6} + 1665972 p^{21} T^{7} + p^{28} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 410628 T - 2165589726190 T^{2} + 6248608032034800 T^{3} + \)\(38\!\cdots\!99\)\( T^{4} + 6248608032034800 p^{7} T^{5} - 2165589726190 p^{14} T^{6} - 410628 p^{21} T^{7} + p^{28} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 1702134 T - 1128927975562 T^{2} - 1618924907272816080 T^{3} + \)\(28\!\cdots\!99\)\( T^{4} - 1618924907272816080 p^{7} T^{5} - 1128927975562 p^{14} T^{6} + 1702134 p^{21} T^{7} + p^{28} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 547526 T - 5309538579326 T^{2} + 370216478913573040 T^{3} + \)\(20\!\cdots\!67\)\( T^{4} + 370216478913573040 p^{7} T^{5} - 5309538579326 p^{14} T^{6} - 547526 p^{21} T^{7} + p^{28} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 2590616 T + 2056450789210 T^{2} + 19343048712604086400 T^{3} - \)\(55\!\cdots\!01\)\( T^{4} + 19343048712604086400 p^{7} T^{5} + 2056450789210 p^{14} T^{6} - 2590616 p^{21} T^{7} + p^{28} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 4129272 T + 22218672158062 T^{2} - 4129272 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 8008868 T + 26118740970730 T^{2} - 1747516196782370000 p T^{3} + \)\(11\!\cdots\!31\)\( p^{2} T^{4} - 1747516196782370000 p^{8} T^{5} + 26118740970730 p^{14} T^{6} - 8008868 p^{21} T^{7} + p^{28} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 2470456 T + 19757905667170 T^{2} - \)\(12\!\cdots\!12\)\( T^{3} - \)\(29\!\cdots\!49\)\( T^{4} - \)\(12\!\cdots\!12\)\( p^{7} T^{5} + 19757905667170 p^{14} T^{6} + 2470456 p^{21} T^{7} + p^{28} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 9900786 T + 68835957963214 T^{2} + 9900786 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 15423492 T + 94566779700986 T^{2} + \)\(84\!\cdots\!40\)\( T^{3} + \)\(80\!\cdots\!87\)\( T^{4} + \)\(84\!\cdots\!40\)\( p^{7} T^{5} + 94566779700986 p^{14} T^{6} + 15423492 p^{21} T^{7} + p^{28} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 17377472 T + 164552259333822 T^{2} + 17377472 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
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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
−7.911138578315997785524343024278, −7.73011276262065396796496846389, −7.18306516783960989989576724160, −6.98177253943191985727691206491, −6.61064088078392944903217079373, −6.59345680504772517635660239053, −6.03601406424306452211692360467, −5.98998670249934286713919032819, −5.47860384921091361982561338109, −5.37504250134008535731227265315, −5.11023333348749384197500922356, −4.51914883626294864979106170958, −4.34434245388672431555624034386, −3.90616793167776405555196925372, −3.82463656962901274900689484969, −3.56223904798319913416939771650, −3.04126622107410103105826048305, −2.63090757917537563341071044358, −2.52949531173661521046451577216, −2.08108131550526648020332241051, −1.62113719499228249659975478702, −1.30449836869852795119954491649, −1.08345390428256736115630164645, −0.56201455436856600575825159913, −0.03769574961545899779633908542,
0.03769574961545899779633908542, 0.56201455436856600575825159913, 1.08345390428256736115630164645, 1.30449836869852795119954491649, 1.62113719499228249659975478702, 2.08108131550526648020332241051, 2.52949531173661521046451577216, 2.63090757917537563341071044358, 3.04126622107410103105826048305, 3.56223904798319913416939771650, 3.82463656962901274900689484969, 3.90616793167776405555196925372, 4.34434245388672431555624034386, 4.51914883626294864979106170958, 5.11023333348749384197500922356, 5.37504250134008535731227265315, 5.47860384921091361982561338109, 5.98998670249934286713919032819, 6.03601406424306452211692360467, 6.59345680504772517635660239053, 6.61064088078392944903217079373, 6.98177253943191985727691206491, 7.18306516783960989989576724160, 7.73011276262065396796496846389, 7.911138578315997785524343024278
