| L(s) = 1 | − 2-s + 3·5-s + 2·7-s − 3·10-s − 6·11-s − 2·14-s − 16-s + 4·17-s + 8·19-s + 6·22-s + 8·23-s + 6·25-s + 2·29-s + 10·31-s − 32-s − 4·34-s + 6·35-s + 20·37-s − 8·38-s + 2·41-s − 12·43-s − 8·46-s + 2·47-s − 9·49-s − 6·50-s − 6·53-s − 18·55-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.34·5-s + 0.755·7-s − 0.948·10-s − 1.80·11-s − 0.534·14-s − 1/4·16-s + 0.970·17-s + 1.83·19-s + 1.27·22-s + 1.66·23-s + 6/5·25-s + 0.371·29-s + 1.79·31-s − 0.176·32-s − 0.685·34-s + 1.01·35-s + 3.28·37-s − 1.29·38-s + 0.312·41-s − 1.82·43-s − 1.17·46-s + 0.291·47-s − 9/7·49-s − 0.848·50-s − 0.824·53-s − 2.42·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.434609170\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.434609170\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) |
| 13 | | \( 1 \) |
| good | 2 | $S_4\times C_2$ | \( 1 + T + T^{2} + T^{3} + p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 2 T + 13 T^{2} - 16 T^{3} + 13 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 6 T + 27 T^{2} + 78 T^{3} + 27 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 4 T + 19 T^{2} - 40 T^{3} + 19 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 8 T + 67 T^{2} - 298 T^{3} + 67 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 8 T + 85 T^{2} - 374 T^{3} + 85 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 2 T + 79 T^{2} - 104 T^{3} + 79 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 10 T + 115 T^{2} - 614 T^{3} + 115 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 20 T + 235 T^{2} - 1708 T^{3} + 235 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 2 T + 79 T^{2} - 92 T^{3} + 79 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 12 T + 141 T^{2} + 1034 T^{3} + 141 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 2 T + 133 T^{2} - 176 T^{3} + 133 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 6 T + 99 T^{2} + 708 T^{3} + 99 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 12 T + 207 T^{2} - 1434 T^{3} + 207 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 6 T + 171 T^{2} + 656 T^{3} + 171 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 18 T + 285 T^{2} - 2520 T^{3} + 285 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 8 T + 223 T^{2} + 1130 T^{3} + 223 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 20 T + 271 T^{2} - 2404 T^{3} + 271 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 12 T + 189 T^{2} - 1864 T^{3} + 189 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 18 T + 333 T^{2} + 3024 T^{3} + 333 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 171 T^{2} - 288 T^{3} + 171 p T^{4} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 14 T + 319 T^{2} + 2692 T^{3} + 319 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−6.95798664613708808027820693714, −6.73171812368204319204339380469, −6.62182511250369655840207349554, −6.22471198212272294022342027389, −6.04294500494271964389574068991, −5.60621613620994947119316024672, −5.59666881766365025936911623501, −5.25887338773235956524320414489, −5.17964885655405421540123643074, −4.98480760923770989490098305146, −4.56077780526234914857742073772, −4.42965787880588545719330136730, −4.34945047894899629539736310235, −3.58780797244453857109727515708, −3.37378592220858612121010657192, −3.16680361381545679195542048337, −2.79522066470066639315538655438, −2.77274157348718618554953751048, −2.28697515783435572310225539325, −2.26880825748809107091976063773, −1.67893364033793827460975273727, −1.32652502207087293819551576700, −0.979648655266459254706469274800, −0.930008591065161958034712359188, −0.39160736299070291462047147226,
0.39160736299070291462047147226, 0.930008591065161958034712359188, 0.979648655266459254706469274800, 1.32652502207087293819551576700, 1.67893364033793827460975273727, 2.26880825748809107091976063773, 2.28697515783435572310225539325, 2.77274157348718618554953751048, 2.79522066470066639315538655438, 3.16680361381545679195542048337, 3.37378592220858612121010657192, 3.58780797244453857109727515708, 4.34945047894899629539736310235, 4.42965787880588545719330136730, 4.56077780526234914857742073772, 4.98480760923770989490098305146, 5.17964885655405421540123643074, 5.25887338773235956524320414489, 5.59666881766365025936911623501, 5.60621613620994947119316024672, 6.04294500494271964389574068991, 6.22471198212272294022342027389, 6.62182511250369655840207349554, 6.73171812368204319204339380469, 6.95798664613708808027820693714
