| L(s) = 1 | − 16·2-s + 64·4-s + 1.02e3·8-s + 3.92e3·9-s + 1.08e4·11-s − 1.63e4·16-s − 6.27e4·18-s − 1.73e5·22-s + 3.31e4·23-s + 1.41e5·25-s − 4.15e5·29-s + 6.55e4·32-s + 2.51e5·36-s + 6.65e5·37-s − 2.60e6·43-s + 6.95e5·44-s − 5.30e5·46-s − 2.26e6·50-s − 1.11e6·53-s + 6.64e6·58-s + 7.86e5·64-s − 7.65e5·67-s − 7.54e6·71-s + 4.01e6·72-s − 1.06e7·74-s − 2.60e6·79-s + 4.78e6·81-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s + 0.707·8-s + 1.79·9-s + 2.46·11-s − 16-s − 2.53·18-s − 3.48·22-s + 0.568·23-s + 1.81·25-s − 3.16·29-s + 0.353·32-s + 0.897·36-s + 2.16·37-s − 4.99·43-s + 1.23·44-s − 0.803·46-s − 2.56·50-s − 1.03·53-s + 4.46·58-s + 3/8·64-s − 0.311·67-s − 2.50·71-s + 1.26·72-s − 3.05·74-s − 0.594·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92236816 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92236816 ^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.987495702\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.987495702\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_2$ | \( ( 1 + p^{3} T + p^{6} T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^3$ | \( 1 - 436 p^{2} T^{2} + 131047 p^{4} T^{4} - 436 p^{16} T^{6} + p^{28} T^{8} \) |
| 5 | $C_2^3$ | \( 1 - 141458 T^{2} + 13906850139 T^{4} - 141458 p^{14} T^{6} + p^{28} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 494 p T + 82985 p^{2} T^{2} - 494 p^{8} T^{3} + p^{14} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 117368522 T^{2} + p^{14} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 815304704 T^{2} + 496343933805126687 T^{4} - 815304704 p^{14} T^{6} + p^{28} T^{8} \) |
| 19 | $C_2^3$ | \( 1 - 1611436836 T^{2} + 1797721990634806775 T^{4} - 1611436836 p^{14} T^{6} + p^{28} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 16580 T - 3129929047 T^{2} - 16580 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 103766 T + p^{7} T^{2} )^{4} \) |
| 31 | $C_2^3$ | \( 1 + 25781595850 T^{2} - 92253250648059097821 T^{4} + 25781595850 p^{14} T^{6} + p^{28} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 - 332798 T + 15822631671 T^{2} - 332798 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 221598586960 T^{2} + p^{14} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 651334 T + p^{7} T^{2} )^{4} \) |
| 47 | $C_2^3$ | \( 1 + 494678271274 T^{2} - \)\(11\!\cdots\!93\)\( T^{4} + 494678271274 p^{14} T^{6} + p^{28} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + 559046 T - 862178709721 T^{2} + 559046 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 4671442429700 T^{2} + \)\(15\!\cdots\!39\)\( T^{4} - 4671442429700 p^{14} T^{6} + p^{28} T^{8} \) |
| 61 | $C_2^3$ | \( 1 + 2907900568158 T^{2} - \)\(14\!\cdots\!77\)\( T^{4} + 2907900568158 p^{14} T^{6} + p^{28} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 + 382884 T - 5914111447867 T^{2} + 382884 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 1886652 T + p^{7} T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 - 2259485255472 T^{2} - \)\(11\!\cdots\!25\)\( T^{4} - 2259485255472 p^{14} T^{6} + p^{28} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 1301660 T - 17509590230559 T^{2} + 1301660 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 37649542741172 T^{2} + p^{14} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - 38536982158880 T^{2} - \)\(47\!\cdots\!41\)\( T^{4} - 38536982158880 p^{14} T^{6} + p^{28} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 + 15655166237184 T^{2} + p^{14} T^{4} )^{2} \) |
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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
−8.677803181566541799721093780541, −8.637693034709180953188644253985, −8.573486580236039555730626021912, −7.966347545902731333984782436235, −7.54734313184641965974610658038, −7.26608347435811285566407424799, −7.25056727840928952461104323100, −6.78115861950891342724138856045, −6.50488710442910020897053474021, −6.41232206223362331345164624177, −5.74645266441989792271384564579, −5.50695942123114402489796305951, −4.77121010600534500294464736864, −4.73236216321193154537712707494, −4.27937646654424869246901921866, −4.13065276117888333288634109406, −3.49013793353478215704969085975, −3.34329680675427887789832248392, −2.90266170458548547068966899923, −1.83104604035566994603810979353, −1.80087400286440759764689842291, −1.41449929016300336040297496401, −1.32224945849858054118684158583, −0.52749968274544998040394169297, −0.47214189922001145519324485762,
0.47214189922001145519324485762, 0.52749968274544998040394169297, 1.32224945849858054118684158583, 1.41449929016300336040297496401, 1.80087400286440759764689842291, 1.83104604035566994603810979353, 2.90266170458548547068966899923, 3.34329680675427887789832248392, 3.49013793353478215704969085975, 4.13065276117888333288634109406, 4.27937646654424869246901921866, 4.73236216321193154537712707494, 4.77121010600534500294464736864, 5.50695942123114402489796305951, 5.74645266441989792271384564579, 6.41232206223362331345164624177, 6.50488710442910020897053474021, 6.78115861950891342724138856045, 7.25056727840928952461104323100, 7.26608347435811285566407424799, 7.54734313184641965974610658038, 7.966347545902731333984782436235, 8.573486580236039555730626021912, 8.637693034709180953188644253985, 8.677803181566541799721093780541
