| L(s) = 1 | + 3·3-s + 3·5-s − 7-s + 6·9-s + 10·11-s − 4·13-s + 9·15-s + 2·17-s + 8·19-s − 3·21-s + 14·23-s + 25-s + 10·27-s − 10·29-s + 3·31-s + 30·33-s − 3·35-s + 15·37-s − 12·39-s + 2·41-s + 6·43-s + 18·45-s − 47-s − 7·49-s + 6·51-s − 17·53-s + 30·55-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 1.34·5-s − 0.377·7-s + 2·9-s + 3.01·11-s − 1.10·13-s + 2.32·15-s + 0.485·17-s + 1.83·19-s − 0.654·21-s + 2.91·23-s + 1/5·25-s + 1.92·27-s − 1.85·29-s + 0.538·31-s + 5.22·33-s − 0.507·35-s + 2.46·37-s − 1.92·39-s + 0.312·41-s + 0.914·43-s + 2.68·45-s − 0.145·47-s − 49-s + 0.840·51-s − 2.33·53-s + 4.04·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 167^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 167^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(29.74706990\) |
| \(L(\frac12)\) |
\(\approx\) |
\(29.74706990\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 167 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 5 | $S_4\times C_2$ | \( 1 - 3 T + 8 T^{2} - 11 T^{3} + 8 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + T + 8 T^{2} - 3 T^{3} + 8 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 10 T + 53 T^{2} - 200 T^{3} + 53 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 4 T + 11 T^{2} + 36 T^{3} + 11 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 2 T + 19 T^{2} - 72 T^{3} + 19 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 8 T + 65 T^{2} - 300 T^{3} + 65 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 14 T + 121 T^{2} - 696 T^{3} + 121 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 10 T + 107 T^{2} + 560 T^{3} + 107 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $C_2$ | \( ( 1 - T + p T^{2} )^{3} \) |
| 37 | $S_4\times C_2$ | \( 1 - 15 T + 176 T^{2} - 1175 T^{3} + 176 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 2 T + 71 T^{2} - 28 T^{3} + 71 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 6 T + 21 T^{2} + 56 T^{3} + 21 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + T + 128 T^{2} + 77 T^{3} + 128 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 17 T + 242 T^{2} + 1891 T^{3} + 242 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 5 T + 102 T^{2} - 525 T^{3} + 102 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 8 T - 9 T^{2} - 688 T^{3} - 9 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 7 T + 2 p T^{2} - 687 T^{3} + 2 p^{2} T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 6 T + 125 T^{2} - 1000 T^{3} + 125 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 20 T + 279 T^{2} + 2780 T^{3} + 279 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 4 T + 189 T^{2} + 664 T^{3} + 189 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 37 T + 672 T^{2} - 7551 T^{3} + 672 p T^{4} - 37 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 7 T + 150 T^{2} - 375 T^{3} + 150 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 5 T + 26 T^{2} - 1065 T^{3} + 26 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.08652195562342196858307620699, −6.47884050000989069704619480812, −6.43275343254887133840160880612, −6.41875003662188611936580787230, −6.03822887634030608508867015201, −5.96722697178826307948596422712, −5.56830726554803868333744167914, −5.01698134309669980778166582563, −4.98039273640560288455928864250, −4.86599968169451553761425036477, −4.61552492693973174824339365057, −4.14658372459425639048294234409, −3.97345816742888616208364746593, −3.53439941560144460111848819168, −3.50190763762931604180974984880, −3.19603404644527493949455262156, −3.09492403690426209251539319901, −2.64566904486375887852464876285, −2.42199041247429393161975977449, −2.12827616363329193267698708441, −1.70262546814340208075824819752, −1.58318777938486975447371674613, −1.19928533516231378532974307369, −0.825265481505541245147233901775, −0.74906466476945230869377300948,
0.74906466476945230869377300948, 0.825265481505541245147233901775, 1.19928533516231378532974307369, 1.58318777938486975447371674613, 1.70262546814340208075824819752, 2.12827616363329193267698708441, 2.42199041247429393161975977449, 2.64566904486375887852464876285, 3.09492403690426209251539319901, 3.19603404644527493949455262156, 3.50190763762931604180974984880, 3.53439941560144460111848819168, 3.97345816742888616208364746593, 4.14658372459425639048294234409, 4.61552492693973174824339365057, 4.86599968169451553761425036477, 4.98039273640560288455928864250, 5.01698134309669980778166582563, 5.56830726554803868333744167914, 5.96722697178826307948596422712, 6.03822887634030608508867015201, 6.41875003662188611936580787230, 6.43275343254887133840160880612, 6.47884050000989069704619480812, 7.08652195562342196858307620699
