| L(s) = 1 | − 2-s + 2·3-s − 4-s − 2·6-s + 3·7-s + 8-s + 9-s + 2·11-s − 2·12-s − 3·13-s − 3·14-s − 16-s − 4·17-s − 18-s − 4·19-s + 6·21-s − 2·22-s − 10·23-s + 2·24-s + 3·26-s − 3·28-s + 24·29-s − 4·31-s + 32-s + 4·33-s + 4·34-s − 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s − 1/2·4-s − 0.816·6-s + 1.13·7-s + 0.353·8-s + 1/3·9-s + 0.603·11-s − 0.577·12-s − 0.832·13-s − 0.801·14-s − 1/4·16-s − 0.970·17-s − 0.235·18-s − 0.917·19-s + 1.30·21-s − 0.426·22-s − 2.08·23-s + 0.408·24-s + 0.588·26-s − 0.566·28-s + 4.45·29-s − 0.718·31-s + 0.176·32-s + 0.696·33-s + 0.685·34-s − 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 7^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 7^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.403008821\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.403008821\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{3} \) |
| 13 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 3 | $S_4\times C_2$ | \( 1 - 2 T + p T^{2} - 4 T^{3} + p^{2} T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 2 T + 27 T^{2} - 36 T^{3} + 27 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 4 T + 41 T^{2} + 140 T^{3} + 41 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 4 T + 58 T^{2} + 148 T^{3} + 58 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 10 T + 70 T^{2} + 324 T^{3} + 70 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 24 T + 272 T^{2} - 1846 T^{3} + 272 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 4 T + 74 T^{2} + 264 T^{3} + 74 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 53 T^{2} + 124 T^{3} + 53 p T^{4} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 2 T + 95 T^{2} - 172 T^{3} + 95 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 10 T + 58 T^{2} + 232 T^{3} + 58 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 8 T + 62 T^{2} - 208 T^{3} + 62 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 8 T + 124 T^{2} + 870 T^{3} + 124 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 4 T + 21 T^{2} - 216 T^{3} + 21 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 67 | $S_4\times C_2$ | \( 1 - 12 T + 77 T^{2} - 632 T^{3} + 77 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 6 T + 191 T^{2} + 868 T^{3} + 191 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 10 T + 120 T^{2} - 1186 T^{3} + 120 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 14 T + 242 T^{2} + 2196 T^{3} + 242 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 12 T - 22 T^{2} + 1276 T^{3} - 22 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 2 T + 172 T^{2} + 66 T^{3} + 172 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 10 T + 320 T^{2} - 1962 T^{3} + 320 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.122561416159479436814178082714, −7.82860591290884897892456858845, −7.69709778193808887446665095622, −7.49147077640232604228675064854, −6.90997058301723654818507128616, −6.66939625279353381482979186062, −6.51941354111180638329477221173, −6.39592604266909862028480660967, −5.80317565623292017931052684358, −5.80051425466418093098220704901, −5.06569972900869203913910110200, −4.97710521023129581095395500029, −4.58490952603226215690408259229, −4.51574009253388545405554850548, −4.21015316889081536562101511777, −4.00921322634448998824147620166, −3.37491041055556279617053478705, −3.27675977183679572257693561225, −2.63696819271160682496151284839, −2.61212740189944162147350397224, −2.11173684236454148072684951355, −1.80622807390784857096023020607, −1.58119057474533452137850714711, −0.73504825217921777704938011322, −0.47157078584385429360182253201,
0.47157078584385429360182253201, 0.73504825217921777704938011322, 1.58119057474533452137850714711, 1.80622807390784857096023020607, 2.11173684236454148072684951355, 2.61212740189944162147350397224, 2.63696819271160682496151284839, 3.27675977183679572257693561225, 3.37491041055556279617053478705, 4.00921322634448998824147620166, 4.21015316889081536562101511777, 4.51574009253388545405554850548, 4.58490952603226215690408259229, 4.97710521023129581095395500029, 5.06569972900869203913910110200, 5.80051425466418093098220704901, 5.80317565623292017931052684358, 6.39592604266909862028480660967, 6.51941354111180638329477221173, 6.66939625279353381482979186062, 6.90997058301723654818507128616, 7.49147077640232604228675064854, 7.69709778193808887446665095622, 7.82860591290884897892456858845, 8.122561416159479436814178082714
