| L(s) = 1 | − 2·3-s + 4-s − 2·7-s + 4·9-s − 6·11-s − 2·12-s + 4·13-s + 4·16-s + 12·17-s − 4·19-s + 4·21-s − 6·23-s − 14·25-s − 4·27-s − 2·28-s + 6·29-s − 4·31-s + 12·33-s + 4·36-s + 14·37-s − 8·39-s − 10·43-s − 6·44-s + 24·47-s − 8·48-s + 49-s − 24·51-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/2·4-s − 0.755·7-s + 4/3·9-s − 1.80·11-s − 0.577·12-s + 1.10·13-s + 16-s + 2.91·17-s − 0.917·19-s + 0.872·21-s − 1.25·23-s − 2.79·25-s − 0.769·27-s − 0.377·28-s + 1.11·29-s − 0.718·31-s + 2.08·33-s + 2/3·36-s + 2.30·37-s − 1.28·39-s − 1.52·43-s − 0.904·44-s + 3.50·47-s − 1.15·48-s + 1/7·49-s − 3.36·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68574961 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68574961 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6422183800\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6422183800\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| good | 2 | $C_2^3$ | \( 1 - T^{2} - 3 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) |
| 3 | $D_4\times C_2$ | \( 1 + 2 T - 4 T^{3} - 5 T^{4} - 4 p T^{5} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2^2$ | \( ( 1 + 7 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 + 6 T + 8 T^{2} + 36 T^{3} + 267 T^{4} + 36 p T^{5} + 8 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 12 T + 77 T^{2} - 396 T^{3} + 1752 T^{4} - 396 p T^{5} + 77 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 6 T - 16 T^{2} + 36 T^{3} + 1347 T^{4} + 36 p T^{5} - 16 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 2 T + 36 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^3$ | \( 1 - 55 T^{2} + 1344 T^{4} - 55 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 10 T + 16 T^{2} - 20 T^{3} + 907 T^{4} - 20 p T^{5} + 16 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 12 T + 82 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 + 6 T + 67 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 18 T + 128 T^{2} + 1404 T^{3} + 15819 T^{4} + 1404 p T^{5} + 128 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 20 T + 205 T^{2} - 1460 T^{3} + 9904 T^{4} - 1460 p T^{5} + 205 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 2 T - 104 T^{2} + 52 T^{3} + 6907 T^{4} + 52 p T^{5} - 104 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 4 T + 123 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 22 T + 252 T^{2} - 22 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - 6 T + 148 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 12 T - 22 T^{2} - 144 T^{3} + 6819 T^{4} - 144 p T^{5} - 22 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 8 T - 38 T^{2} + 736 T^{3} - 5213 T^{4} + 736 p T^{5} - 38 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−10.52807817338731503102535610609, −10.04946838483038308866525894690, −10.04778944953097759754507823259, −9.848580526336600023353608529743, −9.427670507004874679649652828066, −9.253876937203867667498890562736, −8.489321419830709194583118338272, −8.090634438141046613290676055716, −7.924442681759987638474619549944, −7.907149105955935916610388692921, −7.36913073524405301302245032399, −7.36707189308310396421472832899, −6.46336141615095059662223825229, −6.21385964025188172171290512662, −6.20603939431763398514667040850, −5.69536135995479125566446277246, −5.64231538960605410821936467083, −5.02394607171507602611155640516, −4.87308718222571916877411924801, −3.88086092302963128809729415964, −3.79992588545711933676505352967, −3.52957957793634842374956169025, −2.64588382454827270263349499521, −2.20239267430026053557451768584, −1.12534378376491676230474080064,
1.12534378376491676230474080064, 2.20239267430026053557451768584, 2.64588382454827270263349499521, 3.52957957793634842374956169025, 3.79992588545711933676505352967, 3.88086092302963128809729415964, 4.87308718222571916877411924801, 5.02394607171507602611155640516, 5.64231538960605410821936467083, 5.69536135995479125566446277246, 6.20603939431763398514667040850, 6.21385964025188172171290512662, 6.46336141615095059662223825229, 7.36707189308310396421472832899, 7.36913073524405301302245032399, 7.907149105955935916610388692921, 7.924442681759987638474619549944, 8.090634438141046613290676055716, 8.489321419830709194583118338272, 9.253876937203867667498890562736, 9.427670507004874679649652828066, 9.848580526336600023353608529743, 10.04778944953097759754507823259, 10.04946838483038308866525894690, 10.52807817338731503102535610609
