| L(s) = 1 | + 2-s + 3·7-s + 2·8-s − 6·11-s + 6·13-s + 3·14-s + 3·16-s + 6·19-s − 6·22-s + 4·23-s + 6·26-s − 2·29-s + 2·31-s + 3·32-s + 4·37-s + 6·38-s − 2·41-s − 4·43-s + 4·46-s + 8·47-s + 6·49-s + 14·53-s + 6·56-s − 2·58-s − 16·59-s − 6·61-s + 2·62-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.13·7-s + 0.707·8-s − 1.80·11-s + 1.66·13-s + 0.801·14-s + 3/4·16-s + 1.37·19-s − 1.27·22-s + 0.834·23-s + 1.17·26-s − 0.371·29-s + 0.359·31-s + 0.530·32-s + 0.657·37-s + 0.973·38-s − 0.312·41-s − 0.609·43-s + 0.589·46-s + 1.16·47-s + 6/7·49-s + 1.92·53-s + 0.801·56-s − 0.262·58-s − 2.08·59-s − 0.768·61-s + 0.254·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.182875776\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.182875776\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 - T + T^{2} - 3 T^{3} + p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 13 | $S_4\times C_2$ | \( 1 - 6 T + 35 T^{2} - 148 T^{3} + 35 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 35 T^{2} - 16 T^{3} + 35 p T^{4} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 6 T + 53 T^{2} - 188 T^{3} + 53 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 4 T + 61 T^{2} - 168 T^{3} + 61 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 2 T + 35 T^{2} + 76 T^{3} + 35 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 2 T + 41 T^{2} + 60 T^{3} + 41 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 4 T + 31 T^{2} - 232 T^{3} + 31 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 2 T + 63 T^{2} - 36 T^{3} + 63 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 4 T - 15 T^{2} - 488 T^{3} - 15 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 8 T + 109 T^{2} - 624 T^{3} + 109 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 14 T + 171 T^{2} - 1188 T^{3} + 171 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 16 T + 113 T^{2} + 608 T^{3} + 113 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 6 T + 131 T^{2} + 484 T^{3} + 131 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 8 T + 169 T^{2} + 944 T^{3} + 169 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 73 | $S_4\times C_2$ | \( 1 - 18 T + 311 T^{2} - 2732 T^{3} + 311 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 12 T + 221 T^{2} - 1576 T^{3} + 221 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 8 T + 185 T^{2} - 1072 T^{3} + 185 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 14 T + 319 T^{2} + 2532 T^{3} + 319 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 22 T + 255 T^{2} - 2404 T^{3} + 255 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.264654793757776369510632912497, −8.137970056880691523917135844166, −7.76060352040576764076987458162, −7.58215731215526571648830390384, −7.33158688695093308511754128066, −7.26506621643004436877254036969, −6.69813201697403344413262287118, −6.38507741696901287985663120951, −5.95363788676847045588620498288, −5.92061764177142006265864474119, −5.46151810166782578622229690455, −5.19685652811275782803491563851, −5.13888509721662286926770517091, −4.53149678749193540753853653583, −4.52238170939592920252372974136, −4.36704017857835235399520417665, −3.53821637760154305696874352793, −3.53273732417998985577464912587, −3.24523823287756373723275915030, −2.71672537984208023638457113417, −2.41356931023426051398477148174, −1.85670382850295779095377283274, −1.63155352418946846412030406660, −0.864995185033961696016580530337, −0.804343197405415663425383782976,
0.804343197405415663425383782976, 0.864995185033961696016580530337, 1.63155352418946846412030406660, 1.85670382850295779095377283274, 2.41356931023426051398477148174, 2.71672537984208023638457113417, 3.24523823287756373723275915030, 3.53273732417998985577464912587, 3.53821637760154305696874352793, 4.36704017857835235399520417665, 4.52238170939592920252372974136, 4.53149678749193540753853653583, 5.13888509721662286926770517091, 5.19685652811275782803491563851, 5.46151810166782578622229690455, 5.92061764177142006265864474119, 5.95363788676847045588620498288, 6.38507741696901287985663120951, 6.69813201697403344413262287118, 7.26506621643004436877254036969, 7.33158688695093308511754128066, 7.58215731215526571648830390384, 7.76060352040576764076987458162, 8.137970056880691523917135844166, 8.264654793757776369510632912497
