| L(s) = 1 | − 3·2-s − 3·3-s + 6·4-s − 2·5-s + 9·6-s + 3·7-s − 10·8-s + 6·9-s + 6·10-s − 3·11-s − 18·12-s − 9·14-s + 6·15-s + 15·16-s − 7·17-s − 18·18-s − 3·19-s − 12·20-s − 9·21-s + 9·22-s − 7·23-s + 30·24-s − 10·25-s − 10·27-s + 18·28-s − 12·29-s − 18·30-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.73·3-s + 3·4-s − 0.894·5-s + 3.67·6-s + 1.13·7-s − 3.53·8-s + 2·9-s + 1.89·10-s − 0.904·11-s − 5.19·12-s − 2.40·14-s + 1.54·15-s + 15/4·16-s − 1.69·17-s − 4.24·18-s − 0.688·19-s − 2.68·20-s − 1.96·21-s + 1.91·22-s − 1.45·23-s + 6.12·24-s − 2·25-s − 1.92·27-s + 3.40·28-s − 2.22·29-s − 3.28·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 7^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 7^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3872986398\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3872986398\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 3 | $C_1$ | \( ( 1 + T )^{3} \) |
| 7 | $C_1$ | \( ( 1 - T )^{3} \) |
| 13 | | \( 1 \) |
| good | 5 | $A_4\times C_2$ | \( 1 + 2 T + 14 T^{2} + 19 T^{3} + 14 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 + 3 T + 29 T^{2} + 67 T^{3} + 29 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + 7 T + 65 T^{2} + 245 T^{3} + 65 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 + 3 T + 39 T^{2} + 101 T^{3} + 39 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 7 T + 83 T^{2} + 329 T^{3} + 83 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + 12 T + 114 T^{2} + 683 T^{3} + 114 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 + 72 T^{2} - 7 T^{3} + 72 p T^{4} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 - 8 T + 102 T^{2} - 479 T^{3} + 102 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 - 2 T + 80 T^{2} - 37 T^{3} + 80 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + 9 T + 149 T^{2} + 787 T^{3} + 149 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 - 7 T + 148 T^{2} - 651 T^{3} + 148 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 + 152 T^{2} - 7 T^{3} + 152 p T^{4} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 - 2 T + 148 T^{2} - 165 T^{3} + 148 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 - 3 T + 158 T^{2} - 283 T^{3} + 158 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 - 32 T + 526 T^{2} - 5351 T^{3} + 526 p T^{4} - 32 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 6 T + 113 T^{2} + 188 T^{3} + 113 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 13 T + 133 T^{2} - 709 T^{3} + 133 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 - 12 T + 236 T^{2} - 1855 T^{3} + 236 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 7 T + 123 T^{2} + 189 T^{3} + 123 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 + 10 T + 298 T^{2} + 1809 T^{3} + 298 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 - 10 T + 294 T^{2} - 19 p T^{3} + 294 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.25559270490540527029359766238, −6.71036929227936872637472162304, −6.60886452803562823260060312550, −6.51850790934860162605601869789, −6.10895790391267112897609575707, −5.90604861422710567767516032087, −5.69626998492889852549299256364, −5.44801163455606602807500496046, −5.18301064037212026806011823830, −5.11188964812853834471134234842, −4.43489139596415729868279831976, −4.38468957820710088999350848869, −4.26179456586535489128579673293, −3.69372055071814788785623113185, −3.66011137781667334840890740850, −3.53402671260811380479038805376, −2.58938766105828621995092389567, −2.39167573010757554282269512651, −2.33665751353803854453521169501, −1.82200851907099318440130465775, −1.73108018270813567346492154628, −1.53563747636027874423913510635, −0.50923830242280401073080917065, −0.49513059618465809282503911797, −0.47883527559379491527470421676,
0.47883527559379491527470421676, 0.49513059618465809282503911797, 0.50923830242280401073080917065, 1.53563747636027874423913510635, 1.73108018270813567346492154628, 1.82200851907099318440130465775, 2.33665751353803854453521169501, 2.39167573010757554282269512651, 2.58938766105828621995092389567, 3.53402671260811380479038805376, 3.66011137781667334840890740850, 3.69372055071814788785623113185, 4.26179456586535489128579673293, 4.38468957820710088999350848869, 4.43489139596415729868279831976, 5.11188964812853834471134234842, 5.18301064037212026806011823830, 5.44801163455606602807500496046, 5.69626998492889852549299256364, 5.90604861422710567767516032087, 6.10895790391267112897609575707, 6.51850790934860162605601869789, 6.60886452803562823260060312550, 6.71036929227936872637472162304, 7.25559270490540527029359766238
