| L(s) = 1 | + 9·3-s − 13·4-s − 27·5-s + 7·8-s + 54·9-s − 19·11-s − 117·12-s − 39·13-s − 243·15-s + 65·16-s + 25·17-s − 169·19-s + 351·20-s + 28·23-s + 63·24-s + 210·25-s + 270·27-s + 274·29-s + 118·31-s − 182·32-s − 171·33-s − 702·36-s + 151·37-s − 351·39-s − 189·40-s − 66·41-s + 598·43-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 1.62·4-s − 2.41·5-s + 0.309·8-s + 2·9-s − 0.520·11-s − 2.81·12-s − 0.832·13-s − 4.18·15-s + 1.01·16-s + 0.356·17-s − 2.04·19-s + 3.92·20-s + 0.253·23-s + 0.535·24-s + 1.67·25-s + 1.92·27-s + 1.75·29-s + 0.683·31-s − 1.00·32-s − 0.902·33-s − 3.25·36-s + 0.670·37-s − 1.44·39-s − 0.747·40-s − 0.251·41-s + 2.12·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 7^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 7^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - p T )^{3} \) |
| 7 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + p T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 + 13 T^{2} - 7 T^{3} + 13 p^{3} T^{4} + p^{9} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + 27 T + 519 T^{2} + 6399 T^{3} + 519 p^{3} T^{4} + 27 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 19 T + 305 p T^{2} + 52645 T^{3} + 305 p^{4} T^{4} + 19 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 25 T + 13071 T^{2} - 255179 T^{3} + 13071 p^{3} T^{4} - 25 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 169 T + 27040 T^{2} + 2261733 T^{3} + 27040 p^{3} T^{4} + 169 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 28 T + 30890 T^{2} - 575155 T^{3} + 30890 p^{3} T^{4} - 28 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 274 T + 65448 T^{2} - 11935583 T^{3} + 65448 p^{3} T^{4} - 274 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 118 T + 67562 T^{2} - 5423567 T^{3} + 67562 p^{3} T^{4} - 118 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 151 T + 94233 T^{2} - 9299487 T^{3} + 94233 p^{3} T^{4} - 151 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 66 T + 142702 T^{2} + 3205533 T^{3} + 142702 p^{3} T^{4} + 66 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 598 T + 277896 T^{2} - 95440011 T^{3} + 277896 p^{3} T^{4} - 598 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 682 T + 461881 T^{2} + 152251940 T^{3} + 461881 p^{3} T^{4} + 682 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 467 T + 437855 T^{2} + 134146427 T^{3} + 437855 p^{3} T^{4} + 467 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 289 T + 521573 T^{2} + 103645473 T^{3} + 521573 p^{3} T^{4} + 289 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 577 T + 582415 T^{2} + 231450861 T^{3} + 582415 p^{3} T^{4} + 577 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 839 T + 815995 T^{2} - 499210255 T^{3} + 815995 p^{3} T^{4} - 839 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 1039 T + 1009259 T^{2} - 559869309 T^{3} + 1009259 p^{3} T^{4} - 1039 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 434 T + 988104 T^{2} + 344792565 T^{3} + 988104 p^{3} T^{4} + 434 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 1260 T + 1969516 T^{2} + 1298400207 T^{3} + 1969516 p^{3} T^{4} + 1260 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 836 T + 1168832 T^{2} - 997730567 T^{3} + 1168832 p^{3} T^{4} - 836 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 787 T + 1568273 T^{2} - 746234905 T^{3} + 1568273 p^{3} T^{4} - 787 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 267 T + 13485 p T^{2} + 223489775 T^{3} + 13485 p^{4} T^{4} - 267 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−8.313268413348125591236487932211, −7.945875537008783802162652935639, −7.939290217627902791755967295318, −7.61228345819914682326921045614, −7.25711945088504195255389695586, −7.24667566239396755074873593687, −6.64994391370228227028446777720, −6.52496028650423183945692216679, −6.05854077762894332265207377203, −5.90642755771204669575121336611, −5.13195274647103022804235229342, −4.86281682135391241971780410032, −4.83161531806123126514698356735, −4.45021791184645849956630455522, −4.24025494004676902107796678378, −4.21193767495319302942751791508, −3.59481695090179430322308569825, −3.48242419106277427709089910916, −3.40901320613480416128673449252, −2.66167530647812958480823852473, −2.57544354149450041990341637925, −2.22971473322912181315524386849, −1.73135231773274580492302726840, −1.01111129917984652722969318476, −0.939951161569732697132348539497, 0, 0, 0,
0.939951161569732697132348539497, 1.01111129917984652722969318476, 1.73135231773274580492302726840, 2.22971473322912181315524386849, 2.57544354149450041990341637925, 2.66167530647812958480823852473, 3.40901320613480416128673449252, 3.48242419106277427709089910916, 3.59481695090179430322308569825, 4.21193767495319302942751791508, 4.24025494004676902107796678378, 4.45021791184645849956630455522, 4.83161531806123126514698356735, 4.86281682135391241971780410032, 5.13195274647103022804235229342, 5.90642755771204669575121336611, 6.05854077762894332265207377203, 6.52496028650423183945692216679, 6.64994391370228227028446777720, 7.24667566239396755074873593687, 7.25711945088504195255389695586, 7.61228345819914682326921045614, 7.939290217627902791755967295318, 7.945875537008783802162652935639, 8.313268413348125591236487932211
