| L(s) = 1 | + 18·3-s − 36·5-s − 120·7-s + 243·9-s + 200·11-s − 284·13-s − 648·15-s + 2.67e3·17-s − 72·19-s − 2.16e3·21-s + 3.84e3·23-s + 2.65e3·25-s + 2.91e3·27-s − 1.02e4·29-s + 1.04e4·31-s + 3.60e3·33-s + 4.32e3·35-s − 1.31e4·37-s − 5.11e3·39-s + 4.16e3·41-s + 5.83e3·43-s − 8.74e3·45-s + 1.52e3·47-s − 1.48e4·49-s + 4.81e4·51-s − 9.01e3·53-s − 7.20e3·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.643·5-s − 0.925·7-s + 9-s + 0.498·11-s − 0.466·13-s − 0.743·15-s + 2.24·17-s − 0.0457·19-s − 1.06·21-s + 1.51·23-s + 0.850·25-s + 0.769·27-s − 2.25·29-s + 1.96·31-s + 0.575·33-s + 0.596·35-s − 1.57·37-s − 0.538·39-s + 0.386·41-s + 0.481·43-s − 0.643·45-s + 0.100·47-s − 0.885·49-s + 2.59·51-s − 0.440·53-s − 0.320·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.963578611\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.963578611\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 36 T - 1362 T^{2} + 36 p^{5} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 120 T + 29278 T^{2} + 120 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 200 T + 46406 T^{2} - 200 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 284 T + 477054 T^{2} + 284 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2676 T + 4344262 T^{2} - 2676 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 72 T + 4159894 T^{2} + 72 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 3840 T + 9416686 T^{2} - 3840 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 10212 T + 64228638 T^{2} + 10212 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 10488 T + 84559438 T^{2} - 10488 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 13148 T + 125908974 T^{2} + 13148 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4164 T + 216207126 T^{2} - 4164 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 5832 T + 168401542 T^{2} - 5832 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 1520 T + 148144670 T^{2} - 1520 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 9012 T + 816689646 T^{2} + 9012 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 55096 T + 1818478886 T^{2} - 55096 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 63444 T + 2677193342 T^{2} - 63444 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 36792 T + 1148752246 T^{2} + 36792 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 37664 T + 2440628942 T^{2} - 37664 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 37836 T + 2085902966 T^{2} + 37836 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 144888 T + 10711615534 T^{2} - 144888 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 109272 T + 10472055958 T^{2} - 109272 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 32556 T + 8836559958 T^{2} + 32556 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 69092 T + 18339537030 T^{2} - 69092 p^{5} T^{3} + p^{10} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.84777420818175185176547427030, −11.64753137825799725672871975962, −10.77525294123362979969606267420, −10.32037713464577175128723706619, −9.641514428844349198381236481133, −9.573680224997444452517651675826, −8.868269358228176635149130272301, −8.437735989171475934897487820692, −7.67297234337209237225557149368, −7.55513324610459832991885372504, −6.79691935324853731967695139011, −6.42818329765418665761697541823, −5.33347076182725851867081685884, −5.03549561862262983898861989610, −3.88649508300831744123953430522, −3.58792109221384344355500034733, −3.08072357639922092698217801647, −2.38231930350643616385839658314, −1.28602553307221515963530576578, −0.63348728890454893125083650516,
0.63348728890454893125083650516, 1.28602553307221515963530576578, 2.38231930350643616385839658314, 3.08072357639922092698217801647, 3.58792109221384344355500034733, 3.88649508300831744123953430522, 5.03549561862262983898861989610, 5.33347076182725851867081685884, 6.42818329765418665761697541823, 6.79691935324853731967695139011, 7.55513324610459832991885372504, 7.67297234337209237225557149368, 8.437735989171475934897487820692, 8.868269358228176635149130272301, 9.573680224997444452517651675826, 9.641514428844349198381236481133, 10.32037713464577175128723706619, 10.77525294123362979969606267420, 11.64753137825799725672871975962, 11.84777420818175185176547427030
