| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.079800883\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.079800883\) |
| \(L(\frac{3}{2})\) |
|
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
−8.675914945144312149896043017430, −8.199553804307954836910387026616, −8.111493524236244550195408506123, −7.75063598687218310762449694858, −7.54126444756986047731478531029, −7.48026866249807448115919249834, −7.05969336716660629167093953093, −6.59969434494003073522238608640, −6.53278613069888156445633853751, −6.23605127464563992067311999312, −5.74390013451958866340506432230, −5.53159251365768472653656127994, −5.28318340870946592713702623511, −4.66828916005421355636096400508, −4.50859719447397886383825145621, −4.39662646799123025228796828234, −3.85709883086749048254554586483, −3.51121577448641004749618456170, −3.08242217990960402074232689384, −2.69025241550367804267563357782, −2.64656591827992415844184139122, −2.37077552664400798675387097928, −1.62780565985368304645925316415, −0.930081192474243944491813629973, −0.33241759806540986935438631218,
0.33241759806540986935438631218, 0.930081192474243944491813629973, 1.62780565985368304645925316415, 2.37077552664400798675387097928, 2.64656591827992415844184139122, 2.69025241550367804267563357782, 3.08242217990960402074232689384, 3.51121577448641004749618456170, 3.85709883086749048254554586483, 4.39662646799123025228796828234, 4.50859719447397886383825145621, 4.66828916005421355636096400508, 5.28318340870946592713702623511, 5.53159251365768472653656127994, 5.74390013451958866340506432230, 6.23605127464563992067311999312, 6.53278613069888156445633853751, 6.59969434494003073522238608640, 7.05969336716660629167093953093, 7.48026866249807448115919249834, 7.54126444756986047731478531029, 7.75063598687218310762449694858, 8.111493524236244550195408506123, 8.199553804307954836910387026616, 8.675914945144312149896043017430
