| L(s) = 1 | − 3-s + 5-s − 4·7-s + 2·9-s + 5·11-s + 5·13-s − 15-s + 2·17-s + 3·19-s + 4·21-s + 5·23-s − 4·25-s − 27-s − 9·29-s − 5·33-s − 4·35-s − 6·37-s − 5·39-s + 8·41-s − 17·43-s + 2·45-s + 47-s − 2·51-s + 53-s + 5·55-s − 3·57-s + 23·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.51·7-s + 2/3·9-s + 1.50·11-s + 1.38·13-s − 0.258·15-s + 0.485·17-s + 0.688·19-s + 0.872·21-s + 1.04·23-s − 4/5·25-s − 0.192·27-s − 1.67·29-s − 0.870·33-s − 0.676·35-s − 0.986·37-s − 0.800·39-s + 1.24·41-s − 2.59·43-s + 0.298·45-s + 0.145·47-s − 0.280·51-s + 0.137·53-s + 0.674·55-s − 0.397·57-s + 2.99·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28094464 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28094464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.527069054\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.527069054\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $A_4\times C_2$ | \( 1 + T - T^{2} - 2 T^{3} - p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $A_4\times C_2$ | \( 1 - T + p T^{2} - 2 T^{3} + p^{2} T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 + 4 T + 16 T^{2} + 40 T^{3} + 16 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 - 5 T + 31 T^{2} - 102 T^{3} + 31 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $A_4\times C_2$ | \( 1 - 5 T + 37 T^{2} - 122 T^{3} + 37 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 - 2 T + 42 T^{2} - 66 T^{3} + 42 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 - 5 T + 5 T^{2} + 26 T^{3} + 5 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + 9 T + 83 T^{2} + 518 T^{3} + 83 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 41 | $A_4\times C_2$ | \( 1 - 8 T + 103 T^{2} - 528 T^{3} + 103 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + 17 T + 153 T^{2} + 1094 T^{3} + 153 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 - T + 69 T^{2} + 162 T^{3} + 69 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 - T + 25 T^{2} + 150 T^{3} + 25 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 - 23 T + 343 T^{2} - 3090 T^{3} + 343 p T^{4} - 23 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 - 3 T + 155 T^{2} - 274 T^{3} + 155 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + 15 T + 245 T^{2} + 2042 T^{3} + 245 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 - 12 T + 137 T^{2} - 776 T^{3} + 137 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 4 T + 152 T^{2} - 258 T^{3} + 152 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 26 T + 421 T^{2} + 4364 T^{3} + 421 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 - 6 T + 137 T^{2} - 260 T^{3} + 137 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $C_6$ | \( 1 - 18 T + 251 T^{2} - 2180 T^{3} + 251 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 8 T + 271 T^{2} + 1424 T^{3} + 271 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.30229895856585030931639016773, −10.11692880202757560862166080512, −9.869133929594604236173049500670, −9.684624793265342977844469657922, −9.050147139763482269078611405626, −9.015355166181452972281499650357, −8.920216275088990554841838742733, −8.162820140582244309712426365595, −7.946775557002075432635587888010, −7.41470131199694458346438252671, −6.90394247714990305526171331855, −6.81126175373219252355428902360, −6.63093008896007893457608729078, −6.06133450022478352014739267903, −5.84897712143959871445382027343, −5.52399931717913474337243460244, −5.13009513375226831578289402134, −4.57784557938487898207451675032, −3.92591423316113913803295874895, −3.76141859005846777118147382186, −3.34314685729054954977195588438, −3.05986132748281541374303515005, −1.99893753972878975326266874225, −1.56727195113053216494868612301, −0.797885396316369683352032844607,
0.797885396316369683352032844607, 1.56727195113053216494868612301, 1.99893753972878975326266874225, 3.05986132748281541374303515005, 3.34314685729054954977195588438, 3.76141859005846777118147382186, 3.92591423316113913803295874895, 4.57784557938487898207451675032, 5.13009513375226831578289402134, 5.52399931717913474337243460244, 5.84897712143959871445382027343, 6.06133450022478352014739267903, 6.63093008896007893457608729078, 6.81126175373219252355428902360, 6.90394247714990305526171331855, 7.41470131199694458346438252671, 7.946775557002075432635587888010, 8.162820140582244309712426365595, 8.920216275088990554841838742733, 9.015355166181452972281499650357, 9.050147139763482269078611405626, 9.684624793265342977844469657922, 9.869133929594604236173049500670, 10.11692880202757560862166080512, 10.30229895856585030931639016773
