| L(s) = 1 | − 251·4-s − 4.37e3·9-s + 4.66e4·16-s + 1.56e5·25-s + 1.09e6·36-s − 3.29e6·49-s + 1.10e7·61-s − 7.58e6·64-s + 1.43e7·81-s − 3.92e7·100-s + 9.82e7·109-s − 7.79e7·121-s + 127-s + 131-s + 137-s + 139-s − 2.03e8·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.50e8·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 1.96·4-s − 2·9-s + 2.84·16-s + 2·25-s + 3.92·36-s − 4·49-s + 6.26·61-s − 3.61·64-s + 3·81-s − 3.92·100-s + 7.26·109-s − 4·121-s − 5.69·144-s + 4·169-s + 7.84·196-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.707916091\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.707916091\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_2^2$ | \( 1 + 251 T^{2} + p^{14} T^{4} \) |
| 3 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{4} \) |
| 11 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 139543474 T^{2} + p^{14} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 11596 T + p^{7} T^{2} )^{2}( 1 + 11596 T + p^{7} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 3027622786 T^{2} + p^{14} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - 206648 T + p^{7} T^{2} )^{2}( 1 + 206648 T + p^{7} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - 877116235954 T^{2} + p^{14} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 787198687546 T^{2} + p^{14} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2774518 T + p^{7} T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8763536 T + p^{7} T^{2} )^{2}( 1 + 8763536 T + p^{7} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 48542379188534 T^{2} + p^{14} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{4} \) |
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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
−9.557237844480641941667753766612, −9.401118814168460912789238002874, −8.976436655531787016513278355668, −8.534668727229181928560800235950, −8.493018594190836716844041183517, −8.472407308566617002594432602853, −7.83782342769864702549464368284, −7.74069543614472846706975856196, −6.95810426214255529052756717962, −6.75400738454581948133976832256, −6.25974646719341995750109974445, −5.96863326165611249606570110215, −5.42388828964434969242880743269, −5.24508946978280358859880799199, −4.85033649283824192410617345787, −4.77731107790788249505133623392, −4.04371031985421454948512768348, −3.59302730783096012431409730316, −3.34484460754219337254331035915, −2.94756419798967659381662542029, −2.42369642809849450591611463403, −1.78940615354436600649976329241, −1.01523123552933711499117827766, −0.51114354738487845502726211419, −0.45988833920848514253379758868,
0.45988833920848514253379758868, 0.51114354738487845502726211419, 1.01523123552933711499117827766, 1.78940615354436600649976329241, 2.42369642809849450591611463403, 2.94756419798967659381662542029, 3.34484460754219337254331035915, 3.59302730783096012431409730316, 4.04371031985421454948512768348, 4.77731107790788249505133623392, 4.85033649283824192410617345787, 5.24508946978280358859880799199, 5.42388828964434969242880743269, 5.96863326165611249606570110215, 6.25974646719341995750109974445, 6.75400738454581948133976832256, 6.95810426214255529052756717962, 7.74069543614472846706975856196, 7.83782342769864702549464368284, 8.472407308566617002594432602853, 8.493018594190836716844041183517, 8.534668727229181928560800235950, 8.976436655531787016513278355668, 9.401118814168460912789238002874, 9.557237844480641941667753766612
