| L(s) = 1 | + 2-s − 3·3-s − 4-s + 3·5-s − 3·6-s − 3·7-s − 8-s + 6·9-s + 3·10-s − 3·11-s + 3·12-s + 8·13-s − 3·14-s − 9·15-s − 16-s + 12·17-s + 6·18-s + 2·19-s − 3·20-s + 9·21-s − 3·22-s + 2·23-s + 3·24-s + 6·25-s + 8·26-s − 10·27-s + 3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.73·3-s − 1/2·4-s + 1.34·5-s − 1.22·6-s − 1.13·7-s − 0.353·8-s + 2·9-s + 0.948·10-s − 0.904·11-s + 0.866·12-s + 2.21·13-s − 0.801·14-s − 2.32·15-s − 1/4·16-s + 2.91·17-s + 1.41·18-s + 0.458·19-s − 0.670·20-s + 1.96·21-s − 0.639·22-s + 0.417·23-s + 0.612·24-s + 6/5·25-s + 1.56·26-s − 1.92·27-s + 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{3} \cdot 7^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{3} \cdot 7^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.105053131\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.105053131\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) |
| 11 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 - T + p T^{2} - p T^{3} + p^{2} T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 8 T + 53 T^{2} - 204 T^{3} + 53 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 12 T + 92 T^{2} - 26 p T^{3} + 92 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 2 T + 26 T^{2} - 120 T^{3} + 26 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 2 T + 54 T^{2} - 60 T^{3} + 54 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 6 T + 56 T^{2} - 314 T^{3} + 56 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 2 T + 39 T^{2} - 52 T^{3} + 39 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 4 T + 109 T^{2} - 292 T^{3} + 109 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 89 T^{2} + 76 T^{3} + 89 p T^{4} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 8 T + 126 T^{2} + 596 T^{3} + 126 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 6 T + 119 T^{2} - 580 T^{3} + 119 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 16 T + 140 T^{2} - 974 T^{3} + 140 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 8 T + 174 T^{2} + 852 T^{3} + 174 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 36 T^{2} + 578 T^{3} + 36 p T^{4} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 8 T + 93 T^{2} - 224 T^{3} + 93 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 2 T + 119 T^{2} - 68 T^{3} + 119 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 2 T + 203 T^{2} + 276 T^{3} + 203 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 18 T + 287 T^{2} + 2588 T^{3} + 287 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 10 T + 190 T^{2} - 1584 T^{3} + 190 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 2 T - 28 T^{2} + 1106 T^{3} - 28 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 18 T + 308 T^{2} - 2830 T^{3} + 308 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.894149085106993743107113148390, −8.400472573052066830496049984055, −8.260524602104659586100969422671, −7.74771173946090028548696667742, −7.46720777188228196011578225205, −7.18288237160315258489033032213, −6.96517124297829632728909276927, −6.50024684803329558637712668580, −6.18931276150015006518314951753, −5.97548076295552579664459818081, −5.85653637764560989405804031092, −5.57141400112530864868395516464, −5.51938416753831974711773950632, −4.81458666021509121619292655991, −4.71657754385482775448862296121, −4.71509751443552185785746495275, −3.79571584770304697644406656620, −3.58632569765157522957911669834, −3.50634987380567245822901997657, −2.99836757066342688132379713773, −2.58964167009184430126048890280, −1.94068935958519665405785224417, −1.39686093246280100302829021436, −0.811708100176887385856908714038, −0.796909974331526722441096830850,
0.796909974331526722441096830850, 0.811708100176887385856908714038, 1.39686093246280100302829021436, 1.94068935958519665405785224417, 2.58964167009184430126048890280, 2.99836757066342688132379713773, 3.50634987380567245822901997657, 3.58632569765157522957911669834, 3.79571584770304697644406656620, 4.71509751443552185785746495275, 4.71657754385482775448862296121, 4.81458666021509121619292655991, 5.51938416753831974711773950632, 5.57141400112530864868395516464, 5.85653637764560989405804031092, 5.97548076295552579664459818081, 6.18931276150015006518314951753, 6.50024684803329558637712668580, 6.96517124297829632728909276927, 7.18288237160315258489033032213, 7.46720777188228196011578225205, 7.74771173946090028548696667742, 8.260524602104659586100969422671, 8.400472573052066830496049984055, 8.894149085106993743107113148390
