| L(s) = 1 | + 2·2-s − 3·3-s + 2·4-s − 6·6-s + 4·7-s + 8-s + 6·9-s − 8·11-s − 6·12-s + 4·13-s + 8·14-s − 4·16-s + 4·17-s + 12·18-s − 2·19-s − 12·21-s − 16·22-s + 6·23-s − 3·24-s + 25-s + 8·26-s − 10·27-s + 8·28-s + 3·29-s + 6·31-s − 8·32-s + 24·33-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.73·3-s + 4-s − 2.44·6-s + 1.51·7-s + 0.353·8-s + 2·9-s − 2.41·11-s − 1.73·12-s + 1.10·13-s + 2.13·14-s − 16-s + 0.970·17-s + 2.82·18-s − 0.458·19-s − 2.61·21-s − 3.41·22-s + 1.25·23-s − 0.612·24-s + 1/5·25-s + 1.56·26-s − 1.92·27-s + 1.51·28-s + 0.557·29-s + 1.07·31-s − 1.41·32-s + 4.17·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 658503 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 658503 ^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.150627607\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.150627607\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 29 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 - p T + p T^{2} - T^{3} + p^{2} T^{4} - p^{3} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - T^{2} + 8 T^{3} - p T^{4} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 4 T + 20 T^{2} - 48 T^{3} + 20 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 8 T + 48 T^{2} + 180 T^{3} + 48 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 4 T + 32 T^{2} - 6 p T^{3} + 32 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 4 T + 24 T^{2} - 42 T^{3} + 24 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 2 T + 37 T^{2} + 92 T^{3} + 37 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 6 T + 65 T^{2} - 244 T^{3} + 65 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 6 T + 89 T^{2} - 340 T^{3} + 89 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 8 T + 3 p T^{2} - 584 T^{3} + 3 p^{2} T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 2 T + 23 T^{2} + 220 T^{3} + 23 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 4 T + 33 T^{2} + 88 T^{3} + 33 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 12 T + 132 T^{2} + 912 T^{3} + 132 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 8 T + 55 T^{2} - 600 T^{3} + 55 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 20 T + 285 T^{2} + 2472 T^{3} + 285 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 4 T + 167 T^{2} - 432 T^{3} + 167 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 144 T^{2} + 52 T^{3} + 144 p T^{4} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 14 T + 153 T^{2} + 1572 T^{3} + 153 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 8 T + 3 p T^{2} + 1160 T^{3} + 3 p^{2} T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 2 T + 177 T^{2} + 92 T^{3} + 177 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 8 T + 221 T^{2} + 1120 T^{3} + 221 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 8 T + 136 T^{2} + 1350 T^{3} + 136 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 4 T + 219 T^{2} - 880 T^{3} + 219 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−12.91716162531684899506489271394, −12.56347159316272050538248874774, −11.93819997665332638897588759917, −11.70003529826021811099898227533, −11.45223536968715129898319147065, −10.97150262426378419697287322230, −10.91363958242218671181732123404, −10.51410641162841244031060632683, −9.980046809237475541709886375772, −9.833436194454830753690913184198, −8.845876907000719840823775694028, −8.510206370274106808493091110356, −7.922820677970911516553643364350, −7.79068631092226940074225682735, −7.18437778548053477104543419734, −6.62921567822898276532781437031, −6.21860099805657199764978100617, −5.72754420404654707034576687104, −5.42430910976235794358093330333, −4.84531528171055805336140400292, −4.61145723778923099505945585356, −4.52961583187061814094193334521, −3.30630143514452427642507298744, −2.73387108539709217330115468964, −1.52848030486985068602653439662,
1.52848030486985068602653439662, 2.73387108539709217330115468964, 3.30630143514452427642507298744, 4.52961583187061814094193334521, 4.61145723778923099505945585356, 4.84531528171055805336140400292, 5.42430910976235794358093330333, 5.72754420404654707034576687104, 6.21860099805657199764978100617, 6.62921567822898276532781437031, 7.18437778548053477104543419734, 7.79068631092226940074225682735, 7.922820677970911516553643364350, 8.510206370274106808493091110356, 8.845876907000719840823775694028, 9.833436194454830753690913184198, 9.980046809237475541709886375772, 10.51410641162841244031060632683, 10.91363958242218671181732123404, 10.97150262426378419697287322230, 11.45223536968715129898319147065, 11.70003529826021811099898227533, 11.93819997665332638897588759917, 12.56347159316272050538248874774, 12.91716162531684899506489271394
