| L(s) = 1 | − 9·3-s + 3·7-s + 54·9-s − 44·11-s − 13·13-s + 36·17-s − 71·19-s − 27·21-s − 160·23-s − 270·27-s − 184·29-s + 59·31-s + 396·33-s + 350·37-s + 117·39-s + 166·41-s − 341·43-s + 394·47-s − 431·49-s − 324·51-s + 314·53-s + 639·57-s − 538·59-s − 523·61-s + 162·63-s + 543·67-s + 1.44e3·69-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.161·7-s + 2·9-s − 1.20·11-s − 0.277·13-s + 0.513·17-s − 0.857·19-s − 0.280·21-s − 1.45·23-s − 1.92·27-s − 1.17·29-s + 0.341·31-s + 2.08·33-s + 1.55·37-s + 0.480·39-s + 0.632·41-s − 1.20·43-s + 1.22·47-s − 1.25·49-s − 0.889·51-s + 0.813·53-s + 1.48·57-s − 1.18·59-s − 1.09·61-s + 0.323·63-s + 0.990·67-s + 2.51·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 3^{3} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 3^{3} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.823201214\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.823201214\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{3} \) |
| 5 | | \( 1 \) |
| good | 7 | $S_4\times C_2$ | \( 1 - 3 T + 440 T^{2} - 3291 T^{3} + 440 p^{3} T^{4} - 3 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 4 p T + 3337 T^{2} + 92264 T^{3} + 3337 p^{3} T^{4} + 4 p^{7} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + p T + 2370 T^{2} - 62191 T^{3} + 2370 p^{3} T^{4} + p^{7} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 36 T + 5795 T^{2} - 571752 T^{3} + 5795 p^{3} T^{4} - 36 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 71 T + 20156 T^{2} + 971087 T^{3} + 20156 p^{3} T^{4} + 71 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 160 T + 12501 T^{2} + 1193440 T^{3} + 12501 p^{3} T^{4} + 160 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 184 T + 41711 T^{2} + 6989712 T^{3} + 41711 p^{3} T^{4} + 184 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 59 T + 18752 T^{2} - 9239699 T^{3} + 18752 p^{3} T^{4} - 59 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 350 T + 131843 T^{2} - 35233460 T^{3} + 131843 p^{3} T^{4} - 350 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 166 T + 163415 T^{2} - 24342420 T^{3} + 163415 p^{3} T^{4} - 166 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 341 T + 86348 T^{2} + 10962197 T^{3} + 86348 p^{3} T^{4} + 341 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 394 T + 182393 T^{2} - 37306980 T^{3} + 182393 p^{3} T^{4} - 394 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 314 T + 421811 T^{2} - 85610556 T^{3} + 421811 p^{3} T^{4} - 314 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 538 T + 275541 T^{2} + 59557396 T^{3} + 275541 p^{3} T^{4} + 538 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 523 T + 9194 p T^{2} + 231118711 T^{3} + 9194 p^{4} T^{4} + 523 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 543 T + 763644 T^{2} - 254226039 T^{3} + 763644 p^{3} T^{4} - 543 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 910 T + 1222737 T^{2} + 623394700 T^{3} + 1222737 p^{3} T^{4} + 910 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 870 T + 582807 T^{2} - 214665460 T^{3} + 582807 p^{3} T^{4} - 870 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 96 T + 1442381 T^{2} - 90599232 T^{3} + 1442381 p^{3} T^{4} - 96 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 210 T + 1644813 T^{2} + 231374308 T^{3} + 1644813 p^{3} T^{4} + 210 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 64 T + 1631067 T^{2} + 69500032 T^{3} + 1631067 p^{3} T^{4} + 64 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 943 T + 2459630 T^{2} - 1736302651 T^{3} + 2459630 p^{3} T^{4} - 943 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−7.66152532038120347026760459962, −7.20917708499928537050140708666, −7.08822715532891634201450032133, −6.91228381067206980155710842794, −6.28685541890895463278820098716, −6.15473167255748977136910077761, −6.04129987835579334451323205481, −5.90864894826343269098826218303, −5.40391903127241154127524502919, −5.33523657534526663995085260851, −4.83496327855757623202050540668, −4.78677044519512737409724476798, −4.53022494924384189035255843555, −4.04398690482172205349858348073, −3.83612840607792923268670400538, −3.72648294647191813166796082234, −2.91719526339620060340391482866, −2.86575331210029018853654891066, −2.50205181367908111227355371551, −1.88395124264112557922632005411, −1.71383028610398382982922198281, −1.56157093043325149807622077268, −0.66677101232613254400132192256, −0.50240137258061716615414321744, −0.38669029637996410470696547195,
0.38669029637996410470696547195, 0.50240137258061716615414321744, 0.66677101232613254400132192256, 1.56157093043325149807622077268, 1.71383028610398382982922198281, 1.88395124264112557922632005411, 2.50205181367908111227355371551, 2.86575331210029018853654891066, 2.91719526339620060340391482866, 3.72648294647191813166796082234, 3.83612840607792923268670400538, 4.04398690482172205349858348073, 4.53022494924384189035255843555, 4.78677044519512737409724476798, 4.83496327855757623202050540668, 5.33523657534526663995085260851, 5.40391903127241154127524502919, 5.90864894826343269098826218303, 6.04129987835579334451323205481, 6.15473167255748977136910077761, 6.28685541890895463278820098716, 6.91228381067206980155710842794, 7.08822715532891634201450032133, 7.20917708499928537050140708666, 7.66152532038120347026760459962
