| L(s) = 1 | + 3·5-s + 2·7-s − 4·11-s + 2·13-s + 10·19-s − 3·23-s + 6·25-s + 10·31-s + 6·35-s + 2·37-s − 8·41-s + 16·43-s + 4·47-s − 2·49-s − 8·53-s − 12·55-s + 10·61-s + 6·65-s + 10·67-s + 8·71-s + 18·73-s − 8·77-s + 14·79-s − 10·83-s − 2·89-s + 4·91-s + 30·95-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 0.755·7-s − 1.20·11-s + 0.554·13-s + 2.29·19-s − 0.625·23-s + 6/5·25-s + 1.79·31-s + 1.01·35-s + 0.328·37-s − 1.24·41-s + 2.43·43-s + 0.583·47-s − 2/7·49-s − 1.09·53-s − 1.61·55-s + 1.28·61-s + 0.744·65-s + 1.22·67-s + 0.949·71-s + 2.10·73-s − 0.911·77-s + 1.57·79-s − 1.09·83-s − 0.211·89-s + 0.419·91-s + 3.07·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.444018073\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.444018073\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) |
| 23 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 - 2 T + 6 T^{2} - 4 T^{3} + 6 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 4 T + 19 T^{2} + 40 T^{3} + 19 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $D_{6}$ | \( 1 - 2 T + 21 T^{2} - 64 T^{3} + 21 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 30 T^{2} - 18 T^{3} + 30 p T^{4} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 10 T + 71 T^{2} - 328 T^{3} + 71 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 66 T^{2} + 18 T^{3} + 66 p T^{4} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 10 T + 110 T^{2} - 616 T^{3} + 110 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 2 T + 48 T^{2} - 256 T^{3} + 48 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 8 T + 22 T^{2} + 74 T^{3} + 22 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 16 T + 149 T^{2} - 1072 T^{3} + 149 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 4 T + 91 T^{2} - 160 T^{3} + 91 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 8 T + 58 T^{2} + 266 T^{3} + 58 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 156 T^{2} - 18 T^{3} + 156 p T^{4} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 10 T + 161 T^{2} - 928 T^{3} + 161 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 10 T + 218 T^{2} - 1336 T^{3} + 218 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 8 T + 112 T^{2} - 554 T^{3} + 112 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $D_{6}$ | \( 1 - 18 T + 153 T^{2} - 1136 T^{3} + 153 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 14 T + 3 p T^{2} - 2116 T^{3} + 3 p^{2} T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 10 T + 232 T^{2} + 1432 T^{3} + 232 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 2 T + 67 T^{2} - 556 T^{3} + 67 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 20 T + 203 T^{2} + 1544 T^{3} + 203 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.52791916103398432197664279856, −7.40011767727810981457358770620, −6.82168432893485571516960758665, −6.67183457801333938463394652085, −6.35782044232975906956561047434, −6.35328800989069438960007878873, −5.91259818132581918029414095652, −5.57628269647893686768030019779, −5.42435831907342898944353143969, −5.25367606936592739201464919667, −5.01649557363182209801530466337, −4.86598445669859623626863531984, −4.45898978061860136248685448506, −4.03814060687521547228709071863, −3.88790489979371730804556037579, −3.56846794282109298201789725206, −2.94138419891333984158036316766, −2.89498398684534227420128739029, −2.80429404863676946273072101567, −2.18772630102367677477184677524, −1.91798839493946426912604372779, −1.82678201657040856791767364805, −1.01291566206120024120795033286, −0.986783123188129510261269133541, −0.58914337766637151662930508360,
0.58914337766637151662930508360, 0.986783123188129510261269133541, 1.01291566206120024120795033286, 1.82678201657040856791767364805, 1.91798839493946426912604372779, 2.18772630102367677477184677524, 2.80429404863676946273072101567, 2.89498398684534227420128739029, 2.94138419891333984158036316766, 3.56846794282109298201789725206, 3.88790489979371730804556037579, 4.03814060687521547228709071863, 4.45898978061860136248685448506, 4.86598445669859623626863531984, 5.01649557363182209801530466337, 5.25367606936592739201464919667, 5.42435831907342898944353143969, 5.57628269647893686768030019779, 5.91259818132581918029414095652, 6.35328800989069438960007878873, 6.35782044232975906956561047434, 6.67183457801333938463394652085, 6.82168432893485571516960758665, 7.40011767727810981457358770620, 7.52791916103398432197664279856
