| L(s) = 1 | − 3·5-s + 2·11-s + 5·13-s + 2·17-s − 19-s − 6·23-s + 6·25-s − 12·29-s + 3·31-s − 9·37-s − 10·41-s − 43-s − 10·47-s − 4·53-s − 6·55-s + 16·59-s − 2·61-s − 15·65-s − 5·67-s − 20·71-s + 15·73-s − 13·79-s + 2·83-s − 6·85-s − 2·89-s + 3·95-s + 12·97-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.603·11-s + 1.38·13-s + 0.485·17-s − 0.229·19-s − 1.25·23-s + 6/5·25-s − 2.22·29-s + 0.538·31-s − 1.47·37-s − 1.56·41-s − 0.152·43-s − 1.45·47-s − 0.549·53-s − 0.809·55-s + 2.08·59-s − 0.256·61-s − 1.86·65-s − 0.610·67-s − 2.37·71-s + 1.75·73-s − 1.46·79-s + 0.219·83-s − 0.650·85-s − 0.211·89-s + 0.307·95-s + 1.21·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{3} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{3} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) |
| 7 | | \( 1 \) |
| good | 11 | $S_4\times C_2$ | \( 1 - 2 T + 15 T^{2} - 62 T^{3} + 15 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 5 T + 28 T^{2} - 133 T^{3} + 28 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 2 T + 27 T^{2} - 86 T^{3} + 27 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + T + 40 T^{2} + 17 T^{3} + 40 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 6 T + 33 T^{2} + 114 T^{3} + 33 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 12 T + 69 T^{2} + 318 T^{3} + 69 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $C_2$ | \( ( 1 - T + p T^{2} )^{3} \) |
| 37 | $S_4\times C_2$ | \( 1 + 9 T + 90 T^{2} + 475 T^{3} + 90 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 10 T + 45 T^{2} + 46 T^{3} + 45 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + T + 104 T^{2} + p T^{3} + 104 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 10 T + 105 T^{2} + 868 T^{3} + 105 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 4 T + 63 T^{2} + 568 T^{3} + 63 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 16 T + 243 T^{2} - 1906 T^{3} + 243 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 2 T + 167 T^{2} + 248 T^{3} + 167 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 5 T + 190 T^{2} + 673 T^{3} + 190 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 20 T + 327 T^{2} + 3038 T^{3} + 327 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 15 T + 246 T^{2} - 2149 T^{3} + 246 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 13 T + 128 T^{2} + 601 T^{3} + 128 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 2 T + 93 T^{2} - 854 T^{3} + 93 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 2 T + 93 T^{2} + 230 T^{3} + 93 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $D_{6}$ | \( 1 - 12 T + 267 T^{2} - 2212 T^{3} + 267 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.36825270394337514533168206342, −6.91162061184242436850418970171, −6.76329268557715266460838779200, −6.50471220730252947194194981301, −6.31659063927488046482924852093, −6.04251193623633270422292009996, −5.96375100368029268169422457870, −5.42449631077325060868522431512, −5.31501582198471571999853671764, −5.26650942366975393463844671212, −4.71307089275679330842933656711, −4.62566021670843276529538235890, −4.35028462576850072862466286940, −3.80982613714810986485334692932, −3.80323322159947893688801521864, −3.79477823649739910477239591833, −3.43822380176136646210394152842, −3.13745725554918756168225421889, −3.07049874622941792284602314354, −2.31304811257798515757220006014, −2.24552915943317034602214269420, −1.93584355132356114306191711914, −1.38362959492251503507087509907, −1.18765098519370445792183443683, −1.15711403322838043373655295435, 0, 0, 0,
1.15711403322838043373655295435, 1.18765098519370445792183443683, 1.38362959492251503507087509907, 1.93584355132356114306191711914, 2.24552915943317034602214269420, 2.31304811257798515757220006014, 3.07049874622941792284602314354, 3.13745725554918756168225421889, 3.43822380176136646210394152842, 3.79477823649739910477239591833, 3.80323322159947893688801521864, 3.80982613714810986485334692932, 4.35028462576850072862466286940, 4.62566021670843276529538235890, 4.71307089275679330842933656711, 5.26650942366975393463844671212, 5.31501582198471571999853671764, 5.42449631077325060868522431512, 5.96375100368029268169422457870, 6.04251193623633270422292009996, 6.31659063927488046482924852093, 6.50471220730252947194194981301, 6.76329268557715266460838779200, 6.91162061184242436850418970171, 7.36825270394337514533168206342
