| L(s) = 1 | − 3·2-s − 3·3-s + 6·4-s + 9·6-s − 6·7-s − 10·8-s + 6·9-s + 8·11-s − 18·12-s − 10·13-s + 18·14-s + 15·16-s − 2·17-s − 18·18-s − 2·19-s + 18·21-s − 24·22-s − 2·23-s + 30·24-s + 30·26-s − 10·27-s − 36·28-s − 2·29-s − 3·31-s − 21·32-s − 24·33-s + 6·34-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.73·3-s + 3·4-s + 3.67·6-s − 2.26·7-s − 3.53·8-s + 2·9-s + 2.41·11-s − 5.19·12-s − 2.77·13-s + 4.81·14-s + 15/4·16-s − 0.485·17-s − 4.24·18-s − 0.458·19-s + 3.92·21-s − 5.11·22-s − 0.417·23-s + 6.12·24-s + 5.88·26-s − 1.92·27-s − 6.80·28-s − 0.371·29-s − 0.538·31-s − 3.71·32-s − 4.17·33-s + 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 5^{6} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 5^{6} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1207026199\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1207026199\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 3 | $C_1$ | \( ( 1 + T )^{3} \) |
| 5 | | \( 1 \) |
| 31 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 11 | $S_4\times C_2$ | \( 1 - 8 T + 46 T^{2} - 168 T^{3} + 46 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 10 T + 64 T^{2} + 274 T^{3} + 64 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 2 T + 16 T^{2} + 72 T^{3} + 16 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 2 T + 22 T^{2} + 80 T^{3} + 22 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 2 T + 37 T^{2} + 108 T^{3} + 37 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 2 T + 55 T^{2} + 132 T^{3} + 55 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 6 T + 47 T^{2} - 316 T^{3} + 47 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 43 | $S_4\times C_2$ | \( 1 + 53 T^{2} + 16 T^{3} + 53 p T^{4} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 2 T + 106 T^{2} - 192 T^{3} + 106 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 10 T + 115 T^{2} + 1068 T^{3} + 115 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 2 T + 101 T^{2} - 44 T^{3} + 101 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 14 T + 92 T^{2} - 336 T^{3} + 92 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 16 T + 186 T^{2} - 1492 T^{3} + 186 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 8 T + 134 T^{2} + 1156 T^{3} + 134 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 2 T + 87 T^{2} + 100 T^{3} + 87 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 6 T + 230 T^{2} + 920 T^{3} + 230 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 2 T + 66 T^{2} + 576 T^{3} + 66 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 2 T + 123 T^{2} + 36 T^{3} + 123 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 20 T + 416 T^{2} - 4116 T^{3} + 416 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.40292299105762489106555096849, −6.85681711491046444965698395273, −6.85302392465612692636374816777, −6.83359485348203258374259818153, −6.34483667576828508476832043473, −6.33989819461300119027227560027, −6.29572385855561449179275244115, −5.60034557728697662349152055486, −5.54611696313498187488460700140, −5.53520038971741542371087004209, −4.70707677864288064361336871533, −4.57693439784357446219644972495, −4.55487493926639681205846655531, −3.89153613793978506193382659687, −3.71690367204396061211365336465, −3.58507823811645053884986134088, −2.91931212057375102680141612124, −2.84613872603308233384948596307, −2.44900472588570954510904864135, −2.06610244894528914292120562700, −1.60909084102922471626374615863, −1.60293390913198760054675371725, −0.68218171956322224231306452910, −0.66831876971305387295147282183, −0.19583818587752804451851287562,
0.19583818587752804451851287562, 0.66831876971305387295147282183, 0.68218171956322224231306452910, 1.60293390913198760054675371725, 1.60909084102922471626374615863, 2.06610244894528914292120562700, 2.44900472588570954510904864135, 2.84613872603308233384948596307, 2.91931212057375102680141612124, 3.58507823811645053884986134088, 3.71690367204396061211365336465, 3.89153613793978506193382659687, 4.55487493926639681205846655531, 4.57693439784357446219644972495, 4.70707677864288064361336871533, 5.53520038971741542371087004209, 5.54611696313498187488460700140, 5.60034557728697662349152055486, 6.29572385855561449179275244115, 6.33989819461300119027227560027, 6.34483667576828508476832043473, 6.83359485348203258374259818153, 6.85302392465612692636374816777, 6.85681711491046444965698395273, 7.40292299105762489106555096849
