| L(s) = 1 | − 3·5-s + 5·7-s − 2·11-s − 2·17-s + 4·19-s − 7·23-s + 6·25-s + 7·29-s + 16·31-s − 15·35-s + 2·37-s + 41-s − 2·43-s − 13·47-s + 7·49-s + 10·53-s + 6·55-s − 6·59-s − 11·61-s − 67-s − 14·71-s + 16·73-s − 10·77-s + 6·79-s − 21·83-s + 6·85-s + 33·89-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 1.88·7-s − 0.603·11-s − 0.485·17-s + 0.917·19-s − 1.45·23-s + 6/5·25-s + 1.29·29-s + 2.87·31-s − 2.53·35-s + 0.328·37-s + 0.156·41-s − 0.304·43-s − 1.89·47-s + 49-s + 1.37·53-s + 0.809·55-s − 0.781·59-s − 1.40·61-s − 0.122·67-s − 1.66·71-s + 1.87·73-s − 1.13·77-s + 0.675·79-s − 2.30·83-s + 0.650·85-s + 3.49·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.050519634\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.050519634\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 - 5 T + 18 T^{2} - 61 T^{3} + 18 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 2 T + 25 T^{2} + 32 T^{3} + 25 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 15 T^{2} + 36 T^{3} + 15 p T^{4} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 2 T + 15 T^{2} - 40 T^{3} + 15 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 4 T + 53 T^{2} - 148 T^{3} + 53 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 7 T + 18 T^{2} + 19 T^{3} + 18 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 7 T + 82 T^{2} - 379 T^{3} + 82 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 16 T + 169 T^{2} - 1100 T^{3} + 169 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 2 T + 103 T^{2} - 136 T^{3} + 103 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - T + 114 T^{2} - 85 T^{3} + 114 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 2 T + 93 T^{2} + 64 T^{3} + 93 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 13 T + 56 T^{2} + 99 T^{3} + 56 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 10 T + 171 T^{2} - 1036 T^{3} + 171 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 6 T + 117 T^{2} + 780 T^{3} + 117 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 11 T + 142 T^{2} + 823 T^{3} + 142 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + T + 68 T^{2} + 247 T^{3} + 68 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 14 T + 193 T^{2} + 1952 T^{3} + 193 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 16 T + 3 p T^{2} - 1952 T^{3} + 3 p^{2} T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 6 T + 153 T^{2} - 1052 T^{3} + 153 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 21 T + 378 T^{2} + 3729 T^{3} + 378 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{3} \) |
| 97 | $S_4\times C_2$ | \( 1 - 30 T + 447 T^{2} - 4724 T^{3} + 447 p T^{4} - 30 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.32536389848072229252557509874, −6.63223659538818725240047097693, −6.62950702320306926622166290423, −6.60728077523054447225699803564, −6.05963657036066672633582773840, −5.86352316603360306864753078562, −5.83719690812729190284224826266, −5.06908165481595361368492470848, −5.06165728456533685752987019655, −4.90848065314951785125087152897, −4.56975264913423021442671271604, −4.56912637985353994698797481796, −4.30868409656426550620015950016, −3.90668587620570782291715110130, −3.58562439507220593584551691652, −3.42290653579562258051041499232, −2.96540962209912356662064292893, −2.70353089847038432414559472636, −2.67678292134418411711464451573, −2.02931640904530562770810775721, −1.74839899920306359012409309664, −1.63914869269874241208355319344, −0.983502206014866253232267802815, −0.72845727250464624039374310062, −0.41907193708444601593264629393,
0.41907193708444601593264629393, 0.72845727250464624039374310062, 0.983502206014866253232267802815, 1.63914869269874241208355319344, 1.74839899920306359012409309664, 2.02931640904530562770810775721, 2.67678292134418411711464451573, 2.70353089847038432414559472636, 2.96540962209912356662064292893, 3.42290653579562258051041499232, 3.58562439507220593584551691652, 3.90668587620570782291715110130, 4.30868409656426550620015950016, 4.56912637985353994698797481796, 4.56975264913423021442671271604, 4.90848065314951785125087152897, 5.06165728456533685752987019655, 5.06908165481595361368492470848, 5.83719690812729190284224826266, 5.86352316603360306864753078562, 6.05963657036066672633582773840, 6.60728077523054447225699803564, 6.62950702320306926622166290423, 6.63223659538818725240047097693, 7.32536389848072229252557509874
