| L(s) = 1 | + 3·3-s + 3·5-s + 7-s + 6·9-s + 5·11-s − 3·13-s + 9·15-s + 7·17-s + 6·19-s + 3·21-s − 23-s + 6·25-s + 10·27-s + 10·29-s + 2·31-s + 15·33-s + 3·35-s + 11·37-s − 9·39-s − 41-s + 18·45-s − 10·47-s + 21·51-s + 3·53-s + 15·55-s + 18·57-s + 8·59-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 1.34·5-s + 0.377·7-s + 2·9-s + 1.50·11-s − 0.832·13-s + 2.32·15-s + 1.69·17-s + 1.37·19-s + 0.654·21-s − 0.208·23-s + 6/5·25-s + 1.92·27-s + 1.85·29-s + 0.359·31-s + 2.61·33-s + 0.507·35-s + 1.80·37-s − 1.44·39-s − 0.156·41-s + 2.68·45-s − 1.45·47-s + 2.94·51-s + 0.412·53-s + 2.02·55-s + 2.38·57-s + 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 5^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 5^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(14.55954080\) |
| \(L(\frac12)\) |
\(\approx\) |
\(14.55954080\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) |
| 13 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 - T + T^{2} + 26 T^{3} + p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 5 T + 25 T^{2} - 78 T^{3} + 25 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 7 T - T^{2} + 118 T^{3} - p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 6 T + 41 T^{2} - 212 T^{3} + 41 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + T + 49 T^{2} + 6 T^{3} + 49 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 10 T + 75 T^{2} - 396 T^{3} + 75 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 2 T + 29 T^{2} - 156 T^{3} + 29 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 11 T + 135 T^{2} - 794 T^{3} + 135 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + T + 107 T^{2} + 86 T^{3} + 107 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 17 T^{2} + 256 T^{3} + 17 p T^{4} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 10 T + 109 T^{2} + 684 T^{3} + 109 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 3 T + 47 T^{2} - 154 T^{3} + 47 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 8 T + 17 T^{2} + 80 T^{3} + 17 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 3 T + 71 T^{2} + 202 T^{3} + 71 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 12 T + 137 T^{2} + 968 T^{3} + 137 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 19 T + 317 T^{2} + 2858 T^{3} + 317 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 26 T + 399 T^{2} - 4124 T^{3} + 399 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 19 T + 341 T^{2} - 3162 T^{3} + 341 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 4 T + 73 T^{2} + 984 T^{3} + 73 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 13 T + 307 T^{2} - 2334 T^{3} + 307 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 9 T + 135 T^{2} - 326 T^{3} + 135 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.488176953244069843852140818610, −7.958000562754556398094212655808, −7.86087682310024991489931890332, −7.83284790028976280581249569926, −7.41644598289367513556424409482, −6.98649117222764133563811714120, −6.84138808971810097680348953488, −6.39341778376821908333162184638, −6.37236571688298062885456490217, −6.02078879345166859751202156638, −5.44628283922666503759763070940, −5.27280077978339403064666770068, −5.16066591216276798176961603004, −4.50018410272704688600700437853, −4.40225946244392021841634042785, −4.18267650448002378648705745008, −3.43886389330611904716692971374, −3.29129373650755031170999303490, −3.24146399340038109614269410199, −2.62453354036007584172383879859, −2.29421508134718927524124795597, −2.18457329864893026823244378060, −1.31185299239793825359156957105, −1.26170598600573382275781644085, −1.00391181317431803394670358994,
1.00391181317431803394670358994, 1.26170598600573382275781644085, 1.31185299239793825359156957105, 2.18457329864893026823244378060, 2.29421508134718927524124795597, 2.62453354036007584172383879859, 3.24146399340038109614269410199, 3.29129373650755031170999303490, 3.43886389330611904716692971374, 4.18267650448002378648705745008, 4.40225946244392021841634042785, 4.50018410272704688600700437853, 5.16066591216276798176961603004, 5.27280077978339403064666770068, 5.44628283922666503759763070940, 6.02078879345166859751202156638, 6.37236571688298062885456490217, 6.39341778376821908333162184638, 6.84138808971810097680348953488, 6.98649117222764133563811714120, 7.41644598289367513556424409482, 7.83284790028976280581249569926, 7.86087682310024991489931890332, 7.958000562754556398094212655808, 8.488176953244069843852140818610
