| L(s) = 1 | + 3·2-s + 3-s + 6·4-s + 4·5-s + 3·6-s − 3·7-s + 10·8-s − 9-s + 12·10-s + 6·12-s − 2·13-s − 9·14-s + 4·15-s + 15·16-s + 2·17-s − 3·18-s + 3·19-s + 24·20-s − 3·21-s − 9·23-s + 10·24-s + 15·25-s − 6·26-s − 6·27-s − 18·28-s + 5·29-s + 12·30-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 0.577·3-s + 3·4-s + 1.78·5-s + 1.22·6-s − 1.13·7-s + 3.53·8-s − 1/3·9-s + 3.79·10-s + 1.73·12-s − 0.554·13-s − 2.40·14-s + 1.03·15-s + 15/4·16-s + 0.485·17-s − 0.707·18-s + 0.688·19-s + 5.36·20-s − 0.654·21-s − 1.87·23-s + 2.04·24-s + 3·25-s − 1.17·26-s − 1.15·27-s − 3.40·28-s + 0.928·29-s + 2.19·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 11^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 11^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(37.77345807\) |
| \(L(\frac12)\) |
\(\approx\) |
\(37.77345807\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 11 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 - T + 2 T^{2} + p T^{3} + 2 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - 4 T + T^{2} + 14 T^{3} + p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 3 T + 16 T^{2} + 39 T^{3} + 16 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 2 T + 19 T^{2} + 60 T^{3} + 19 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 2 T - 5 T^{2} + 128 T^{3} - 5 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 9 T + 64 T^{2} + 377 T^{3} + 64 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 5 T + 70 T^{2} - 287 T^{3} + 70 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 37 | $S_4\times C_2$ | \( 1 - 19 T + 224 T^{2} - 1607 T^{3} + 224 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 12 T + 3 p T^{2} + 782 T^{3} + 3 p^{2} T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 8 T + 81 T^{2} - 400 T^{3} + 81 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - T + 124 T^{2} - 73 T^{3} + 124 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 7 T + 42 T^{2} - 101 T^{3} + 42 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 19 T + 186 T^{2} - 1351 T^{3} + 186 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 2 T + 125 T^{2} + 262 T^{3} + 125 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{3} \) |
| 71 | $S_4\times C_2$ | \( 1 - 8 T + 93 T^{2} - 470 T^{3} + 93 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 26 T + 427 T^{2} + 4312 T^{3} + 427 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 24 T + 381 T^{2} - 3994 T^{3} + 381 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 6 T + 223 T^{2} - 1014 T^{3} + 223 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 8 T + 179 T^{2} - 1202 T^{3} + 179 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 2 T + 55 T^{2} + 228 T^{3} + 55 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.03750946580691331089001568840, −6.93115505190474380175688523164, −6.72208762643286914001392118485, −6.61727861830613984056475578108, −6.13210659523476412886912245614, −6.08784277224706502313266702812, −6.00285903753725849462907430345, −5.47374229478189284974751147138, −5.44448390536990787620306068464, −5.26757836078792562704416603642, −4.76788683998718972843084480770, −4.75663219899512560125587348929, −4.28434205170875391343541615601, −4.07601852975192026102825684353, −3.67362772938905510400006124968, −3.49480851783641473123087468835, −3.30246082963627593562985245868, −2.73265180427454296010832654275, −2.71040547585385724330694653090, −2.39088755802454320538277123851, −2.30633759445980750002686321381, −1.91189434212697828199463691300, −1.43597856281681270903327205740, −0.907887052227383279360662551019, −0.66670168236942445459429456296,
0.66670168236942445459429456296, 0.907887052227383279360662551019, 1.43597856281681270903327205740, 1.91189434212697828199463691300, 2.30633759445980750002686321381, 2.39088755802454320538277123851, 2.71040547585385724330694653090, 2.73265180427454296010832654275, 3.30246082963627593562985245868, 3.49480851783641473123087468835, 3.67362772938905510400006124968, 4.07601852975192026102825684353, 4.28434205170875391343541615601, 4.75663219899512560125587348929, 4.76788683998718972843084480770, 5.26757836078792562704416603642, 5.44448390536990787620306068464, 5.47374229478189284974751147138, 6.00285903753725849462907430345, 6.08784277224706502313266702812, 6.13210659523476412886912245614, 6.61727861830613984056475578108, 6.72208762643286914001392118485, 6.93115505190474380175688523164, 7.03750946580691331089001568840
