| L(s) = 1 | − 3·3-s + 3·5-s − 2·7-s + 6·9-s + 11-s + 7·13-s − 9·15-s + 3·17-s − 19-s + 6·21-s − 3·23-s − 2·25-s − 10·27-s + 10·29-s − 10·31-s − 3·33-s − 6·35-s + 8·37-s − 21·39-s + 41-s + 5·43-s + 18·45-s + 4·47-s − 49-s − 9·51-s + 12·53-s + 3·55-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1.34·5-s − 0.755·7-s + 2·9-s + 0.301·11-s + 1.94·13-s − 2.32·15-s + 0.727·17-s − 0.229·19-s + 1.30·21-s − 0.625·23-s − 2/5·25-s − 1.92·27-s + 1.85·29-s − 1.79·31-s − 0.522·33-s − 1.01·35-s + 1.31·37-s − 3.36·39-s + 0.156·41-s + 0.762·43-s + 2.68·45-s + 0.583·47-s − 1/7·49-s − 1.26·51-s + 1.64·53-s + 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 3^{3} \cdot 17^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 3^{3} \cdot 17^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.442529754\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.442529754\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 17 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 5 | $S_4\times C_2$ | \( 1 - 3 T + 11 T^{2} - 26 T^{3} + 11 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 2 T + 5 T^{2} + 12 T^{3} + 5 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - T + 17 T^{2} - 38 T^{3} + 17 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 7 T + 3 p T^{2} - 178 T^{3} + 3 p^{2} T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + T + 41 T^{2} + 54 T^{3} + 41 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 3 T + 65 T^{2} + 130 T^{3} + 65 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 10 T + 59 T^{2} - 236 T^{3} + 59 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 10 T + 29 T^{2} - 36 T^{3} + 29 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 8 T + 115 T^{2} - 560 T^{3} + 115 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - T + 67 T^{2} + 90 T^{3} + 67 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 5 T + 105 T^{2} - 446 T^{3} + 105 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 4 T + 77 T^{2} - 248 T^{3} + 77 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 12 T + 179 T^{2} - 1208 T^{3} + 179 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 18 T + 257 T^{2} - 2156 T^{3} + 257 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 10 T + 187 T^{2} - 1132 T^{3} + 187 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 4 T + 89 T^{2} - 600 T^{3} + 89 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 4 T + 157 T^{2} + 312 T^{3} + 157 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 4 T + 159 T^{2} - 328 T^{3} + 159 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 6 T + 93 T^{2} + 516 T^{3} + 93 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 20 T + 137 T^{2} - 568 T^{3} + 137 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 6 T + 167 T^{2} + 724 T^{3} + 167 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 26 T + 447 T^{2} - 5132 T^{3} + 447 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.594015502636873465158281821512, −7.942431518440154927691131929358, −7.75851013819455600339650013143, −7.47943411447964275621539690898, −7.13947454856216394228161297581, −6.74375000903032064334040761224, −6.60452074660752177535631638824, −6.24873559705433383046865173703, −6.16306778987565442368096537443, −5.91068392849339819467755840037, −5.67257651735996096288572285333, −5.39392241753206658975643861617, −5.24000959711364024903925359218, −4.83080084139464838887104804343, −4.29227166477465897982427676619, −4.09109046053841299230864449990, −3.73872720323261031582607913637, −3.63152726020550958510695039269, −3.15110185021728928535561738674, −2.42140397895564582618677977968, −2.28889447086047333377323055724, −1.88820488620305467050661342958, −1.19821336359186939643948134767, −1.02075818104333479442614380884, −0.54333767801081341089932029220,
0.54333767801081341089932029220, 1.02075818104333479442614380884, 1.19821336359186939643948134767, 1.88820488620305467050661342958, 2.28889447086047333377323055724, 2.42140397895564582618677977968, 3.15110185021728928535561738674, 3.63152726020550958510695039269, 3.73872720323261031582607913637, 4.09109046053841299230864449990, 4.29227166477465897982427676619, 4.83080084139464838887104804343, 5.24000959711364024903925359218, 5.39392241753206658975643861617, 5.67257651735996096288572285333, 5.91068392849339819467755840037, 6.16306778987565442368096537443, 6.24873559705433383046865173703, 6.60452074660752177535631638824, 6.74375000903032064334040761224, 7.13947454856216394228161297581, 7.47943411447964275621539690898, 7.75851013819455600339650013143, 7.942431518440154927691131929358, 8.594015502636873465158281821512
