| L(s) = 1 | + 3·5-s − 3·7-s + 2·8-s − 12·19-s + 6·25-s − 6·29-s − 3·31-s − 9·35-s − 6·37-s + 6·40-s + 9·43-s − 12·47-s − 9·49-s − 6·56-s − 6·59-s − 3·61-s − 4·64-s − 9·67-s − 12·71-s − 21·73-s − 3·79-s − 18·83-s + 30·89-s − 36·95-s − 15·97-s + 24·101-s + 9·103-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 1.13·7-s + 0.707·8-s − 2.75·19-s + 6/5·25-s − 1.11·29-s − 0.538·31-s − 1.52·35-s − 0.986·37-s + 0.948·40-s + 1.37·43-s − 1.75·47-s − 9/7·49-s − 0.801·56-s − 0.781·59-s − 0.384·61-s − 1/2·64-s − 1.09·67-s − 1.42·71-s − 2.45·73-s − 0.337·79-s − 1.97·83-s + 3.17·89-s − 3.69·95-s − 1.52·97-s + 2.38·101-s + 0.886·103-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $D_{6}$ | \( 1 - p T^{3} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 3 T + 18 T^{2} + 39 T^{3} + 18 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 9 T^{2} + 16 T^{3} + 9 p T^{4} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 9 T^{2} + 98 T^{3} + 9 p T^{4} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 12 T + 93 T^{2} + 460 T^{3} + 93 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 21 T^{2} - 96 T^{3} + 21 p T^{4} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 6 T + 3 p T^{2} + 334 T^{3} + 3 p^{2} T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 3 T - 177 T^{3} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 6 T + 75 T^{2} + 452 T^{3} + 75 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 99 T^{2} + 26 T^{3} + 99 p T^{4} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 9 T + 144 T^{2} - 763 T^{3} + 144 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 12 T + 135 T^{2} + 922 T^{3} + 135 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 135 T^{2} - 16 T^{3} + 135 p T^{4} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 6 T + 165 T^{2} + 694 T^{3} + 165 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 3 T + 162 T^{2} + 299 T^{3} + 162 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 9 T + 120 T^{2} + 855 T^{3} + 120 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 12 T + 165 T^{2} + 1022 T^{3} + 165 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 21 T + 312 T^{2} + 2977 T^{3} + 312 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 3 T + 144 T^{2} + 111 T^{3} + 144 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 18 T + 309 T^{2} + 2820 T^{3} + 309 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 30 T + 555 T^{2} - 6222 T^{3} + 555 p T^{4} - 30 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 15 T + 282 T^{2} + 2321 T^{3} + 282 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.26751085836695021815108974995, −7.04910265352168294128889649272, −6.72448111164943452419810166337, −6.47198296561747758070656703590, −6.34252055364059662500374024090, −6.31339064940290526172002092441, −5.90584262395930892676214656707, −5.76111336381996644627705596445, −5.57872573533584207235905825244, −5.18726328580393753610357782212, −4.82976939833189375859134873866, −4.66827465519160561276170367934, −4.66055279659388285275401737230, −3.99627831995318337721132492919, −3.95718905851133141008080033123, −3.87826655564846433774822227088, −3.19527796236239245220368452352, −3.16371099255121964907188189966, −2.78783314704637761110866440851, −2.59359193434862546954718694390, −2.17749890594373206039450747732, −1.90147249794719231574300785471, −1.65912598273195686657349874909, −1.40428884928330874270402697359, −1.14176125531208770803233411891, 0, 0, 0,
1.14176125531208770803233411891, 1.40428884928330874270402697359, 1.65912598273195686657349874909, 1.90147249794719231574300785471, 2.17749890594373206039450747732, 2.59359193434862546954718694390, 2.78783314704637761110866440851, 3.16371099255121964907188189966, 3.19527796236239245220368452352, 3.87826655564846433774822227088, 3.95718905851133141008080033123, 3.99627831995318337721132492919, 4.66055279659388285275401737230, 4.66827465519160561276170367934, 4.82976939833189375859134873866, 5.18726328580393753610357782212, 5.57872573533584207235905825244, 5.76111336381996644627705596445, 5.90584262395930892676214656707, 6.31339064940290526172002092441, 6.34252055364059662500374024090, 6.47198296561747758070656703590, 6.72448111164943452419810166337, 7.04910265352168294128889649272, 7.26751085836695021815108974995
