| L(s) = 1 | − 2-s + 4·3-s − 3·4-s − 4·6-s + 4·8-s + 4·9-s − 12·12-s + 11·13-s + 3·16-s + 9·17-s − 4·18-s + 10·19-s + 23-s + 16·24-s − 8·25-s − 11·26-s − 9·27-s + 6·29-s + 20·31-s − 6·32-s − 9·34-s − 12·36-s − 4·37-s − 10·38-s + 44·39-s + 21·41-s + 2·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 2.30·3-s − 3/2·4-s − 1.63·6-s + 1.41·8-s + 4/3·9-s − 3.46·12-s + 3.05·13-s + 3/4·16-s + 2.18·17-s − 0.942·18-s + 2.29·19-s + 0.208·23-s + 3.26·24-s − 8/5·25-s − 2.15·26-s − 1.73·27-s + 1.11·29-s + 3.59·31-s − 1.06·32-s − 1.54·34-s − 2·36-s − 0.657·37-s − 1.62·38-s + 7.04·39-s + 3.27·41-s + 0.304·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 47^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 47^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.441252631\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.441252631\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 47 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 2 | $A_4\times C_2$ | \( 1 + T + p^{2} T^{2} + 3 T^{3} + p^{3} T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 3 | $A_4\times C_2$ | \( 1 - 4 T + 4 p T^{2} - 23 T^{3} + 4 p^{2} T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $A_4\times C_2$ | \( 1 + 8 T^{2} + 7 T^{3} + 8 p T^{4} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 + 12 T^{2} - 7 T^{3} + 12 p T^{4} + p^{3} T^{6} \) |
| 13 | $A_4\times C_2$ | \( 1 - 11 T + 77 T^{2} - 327 T^{3} + 77 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 - 9 T + 71 T^{2} - 319 T^{3} + 71 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 - 10 T + 88 T^{2} - 409 T^{3} + 88 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 - T + 25 T^{2} + 81 T^{3} + 25 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 - 6 T + 78 T^{2} - 321 T^{3} + 78 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 - 20 T + 224 T^{2} - 1521 T^{3} + 224 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 4 T - 5 T^{2} - 272 T^{3} - 5 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 - 21 T + 242 T^{2} - 1813 T^{3} + 242 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 - 2 T + 9 T^{2} + 4 p T^{3} + 9 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 - 10 T + 127 T^{2} - 732 T^{3} + 127 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 - 5 T + 155 T^{2} - 577 T^{3} + 155 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + T + 83 T^{2} + 303 T^{3} + 83 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 - 14 T + 110 T^{2} - 693 T^{3} + 110 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 13 T + 225 T^{2} + 1833 T^{3} + 225 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 11 T + 215 T^{2} - 1605 T^{3} + 215 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{3} \) |
| 83 | $A_4\times C_2$ | \( 1 + 7 T + 221 T^{2} + 959 T^{3} + 221 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 + 5 T + 105 T^{2} + 1017 T^{3} + 105 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 - 17 T + 322 T^{2} - 3257 T^{3} + 322 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.295482710556109897228732489522, −7.75504710342140520468789070923, −7.69071080447590782122298674540, −7.57572696377857331893156243985, −7.35996037965023923998538845108, −6.70901771519397340418789446236, −6.34010615058681317371997227473, −6.07593344385307373523405363413, −5.78851357418825226441019183477, −5.77871235067677181626652134387, −5.29355471509519954914317535587, −5.05641735129151060254190309995, −4.64377601628690343807144936306, −4.04617174907023131121516301223, −4.03764894447960354163739878408, −3.95714571417728861745957841194, −3.34752105717948397321166279371, −3.19448840492233493025733420146, −3.03542812218951055302663031021, −2.66072166399865095649191348474, −2.38035721359749124554264776031, −1.64876158354136339895050115268, −1.13335330144779716081618375989, −0.931708563042401525700774881660, −0.78202431895276622181323180664,
0.78202431895276622181323180664, 0.931708563042401525700774881660, 1.13335330144779716081618375989, 1.64876158354136339895050115268, 2.38035721359749124554264776031, 2.66072166399865095649191348474, 3.03542812218951055302663031021, 3.19448840492233493025733420146, 3.34752105717948397321166279371, 3.95714571417728861745957841194, 4.03764894447960354163739878408, 4.04617174907023131121516301223, 4.64377601628690343807144936306, 5.05641735129151060254190309995, 5.29355471509519954914317535587, 5.77871235067677181626652134387, 5.78851357418825226441019183477, 6.07593344385307373523405363413, 6.34010615058681317371997227473, 6.70901771519397340418789446236, 7.35996037965023923998538845108, 7.57572696377857331893156243985, 7.69071080447590782122298674540, 7.75504710342140520468789070923, 8.295482710556109897228732489522
