| L(s) = 1 | − 104·4-s − 520·7-s − 1.37e4·13-s + 1.63e4·16-s + 1.32e5·19-s + 6.40e4·25-s + 5.40e4·28-s − 3.01e3·31-s − 1.52e6·37-s − 1.52e4·43-s + 1.71e6·49-s + 1.43e6·52-s + 1.97e6·61-s − 3.98e6·64-s − 7.71e6·67-s − 8.01e6·73-s − 1.38e7·76-s − 5.39e6·79-s + 7.16e6·91-s + 2.59e7·97-s − 6.66e6·100-s − 1.01e7·103-s + 2.63e7·109-s − 8.51e6·112-s + 2.11e6·121-s + 3.13e5·124-s + 127-s + ⋯ |
| L(s) = 1 | − 0.812·4-s − 0.573·7-s − 1.73·13-s + 16-s + 4.43·19-s + 0.820·25-s + 0.465·28-s − 0.0181·31-s − 4.94·37-s − 0.0293·43-s + 2.08·49-s + 1.41·52-s + 1.11·61-s − 1.90·64-s − 3.13·67-s − 2.41·73-s − 3.60·76-s − 1.23·79-s + 0.996·91-s + 2.88·97-s − 0.666·100-s − 0.914·103-s + 1.95·109-s − 0.573·112-s + 0.108·121-s + 0.0147·124-s − 2.54·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43046721 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43046721 ^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.9114657509\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9114657509\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 + 13 p^{3} T^{2} - 87 p^{6} T^{4} + 13 p^{17} T^{6} + p^{28} T^{8} \) |
| 5 | $C_2^3$ | \( 1 - 12818 p T^{2} - 79839501 p^{2} T^{4} - 12818 p^{15} T^{6} + p^{28} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 + 260 T - 755943 T^{2} + 260 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 - 2110342 T^{2} - 375296290226277 T^{4} - 2110342 p^{14} T^{6} + p^{28} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 530 p T - 90393 p^{2} T^{2} + 530 p^{8} T^{3} + p^{14} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 259975906 T^{2} + p^{14} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 33176 T + p^{7} T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 - 5812848334 T^{2} + 22196369429547825747 T^{4} - 5812848334 p^{14} T^{6} + p^{28} T^{8} \) |
| 29 | $C_2^3$ | \( 1 - 15420328618 T^{2} - 59771697988689673557 T^{4} - 15420328618 p^{14} T^{6} + p^{28} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 + 1508 T - 27510340047 T^{2} + 1508 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 380770 T + p^{7} T^{2} )^{4} \) |
| 41 | $C_2^3$ | \( 1 - 381757891762 T^{2} + \)\(10\!\cdots\!83\)\( T^{4} - 381757891762 p^{14} T^{6} + p^{28} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 + 7640 T - 271760241507 T^{2} + 7640 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 693036689566 T^{2} + \)\(22\!\cdots\!87\)\( T^{4} - 693036689566 p^{14} T^{6} + p^{28} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + 1288434979834 T^{2} + p^{14} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 + 2355536454362 T^{2} - \)\(64\!\cdots\!17\)\( T^{4} + 2355536454362 p^{14} T^{6} + p^{28} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 988858 T - 2164902691857 T^{2} - 988858 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 3857360 T + 8818514564277 T^{2} + 3857360 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 332728892782 T^{2} + p^{14} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2004730 T + p^{7} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 2699684 T - 11915615286303 T^{2} + 2699684 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 46919519671414 T^{2} + \)\(14\!\cdots\!67\)\( T^{4} - 46919519671414 p^{14} T^{6} + p^{28} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 28535629791058 T^{2} + p^{14} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 12957490 T + 87098262621987 T^{2} - 12957490 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
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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
−9.074574578635404174491299855660, −8.826039303224072961920359976597, −8.723433083663871635676578078648, −8.448815165509423933782477050791, −7.56582832016604216500301744223, −7.49867813951981142662348787813, −7.38121483296721386668263917185, −7.20980635756254692348717300876, −6.85030227278402328626250081780, −6.25786479334288271828640722147, −5.69387287409055102442576408621, −5.52107163731535044454189489772, −5.26493325220441662393298707363, −5.09536761808291883322496851145, −4.57502619769594332548424611794, −4.29307893444270943279501648733, −3.62061222796596329514629770093, −3.11390947818468272796833128664, −3.07444275623438028772978185273, −3.03893653861249466631195688457, −1.89129503046024634684032388027, −1.72492735574352251411899384415, −1.02853509131055663135292752054, −0.75554940101504155263071341128, −0.17945035113760507038175970496,
0.17945035113760507038175970496, 0.75554940101504155263071341128, 1.02853509131055663135292752054, 1.72492735574352251411899384415, 1.89129503046024634684032388027, 3.03893653861249466631195688457, 3.07444275623438028772978185273, 3.11390947818468272796833128664, 3.62061222796596329514629770093, 4.29307893444270943279501648733, 4.57502619769594332548424611794, 5.09536761808291883322496851145, 5.26493325220441662393298707363, 5.52107163731535044454189489772, 5.69387287409055102442576408621, 6.25786479334288271828640722147, 6.85030227278402328626250081780, 7.20980635756254692348717300876, 7.38121483296721386668263917185, 7.49867813951981142662348787813, 7.56582832016604216500301744223, 8.448815165509423933782477050791, 8.723433083663871635676578078648, 8.826039303224072961920359976597, 9.074574578635404174491299855660
