| L(s) = 1 | + 81·3-s + 351·7-s + 4.37e3·9-s + 3.13e3·11-s + 3.58e3·13-s + 5.22e3·17-s + 2.11e4·19-s + 2.84e4·21-s + 4.04e4·23-s + 1.96e5·27-s + 2.65e4·29-s − 1.43e5·31-s + 2.54e5·33-s + 5.36e5·37-s + 2.90e5·39-s − 4.81e5·41-s − 2.21e5·43-s + 1.80e6·47-s − 3.28e5·49-s + 4.23e5·51-s + 3.90e5·53-s + 1.71e6·57-s + 1.99e6·59-s + 1.07e5·61-s + 1.53e6·63-s + 3.71e6·67-s + 3.27e6·69-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.386·7-s + 2·9-s + 0.710·11-s + 0.452·13-s + 0.257·17-s + 0.708·19-s + 0.669·21-s + 0.692·23-s + 1.92·27-s + 0.202·29-s − 0.864·31-s + 1.23·33-s + 1.74·37-s + 0.783·39-s − 1.09·41-s − 0.423·43-s + 2.53·47-s − 0.398·49-s + 0.446·51-s + 0.360·53-s + 1.22·57-s + 1.26·59-s + 0.0607·61-s + 0.773·63-s + 1.50·67-s + 1.20·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27000000 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27000000 ^{s/2} \, \Gamma_{\C}(s+7/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(21.91570773\) |
| \(L(\frac12)\) |
\(\approx\) |
\(21.91570773\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{3} T )^{3} \) |
| 5 | | \( 1 \) |
| good | 7 | $S_4\times C_2$ | \( 1 - 351 T + 451296 T^{2} - 1241701999 T^{3} + 451296 p^{7} T^{4} - 351 p^{14} T^{5} + p^{21} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 3138 T - 7146939 T^{2} + 165372278268 T^{3} - 7146939 p^{7} T^{4} - 3138 p^{14} T^{5} + p^{21} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 3585 T + 63520026 T^{2} + 262165915235 T^{3} + 63520026 p^{7} T^{4} - 3585 p^{14} T^{5} + p^{21} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 5226 T + 1062048111 T^{2} - 4239709613484 T^{3} + 1062048111 p^{7} T^{4} - 5226 p^{14} T^{5} + p^{21} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 21195 T + 2440310292 T^{2} - 36476915881835 T^{3} + 2440310292 p^{7} T^{4} - 21195 p^{14} T^{5} + p^{21} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 1758 p T + 6810964593 T^{2} - 316626867509748 T^{3} + 6810964593 p^{7} T^{4} - 1758 p^{15} T^{5} + p^{21} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 26598 T + 36888283995 T^{2} - 144059349906660 T^{3} + 36888283995 p^{7} T^{4} - 26598 p^{14} T^{5} + p^{21} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 143463 T + 34387481256 T^{2} + 6306882735108247 T^{3} + 34387481256 p^{7} T^{4} + 143463 p^{14} T^{5} + p^{21} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 536658 T + 327691851987 T^{2} - 2774048395900732 p T^{3} + 327691851987 p^{7} T^{4} - 536658 p^{14} T^{5} + p^{21} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 481320 T + 118094246043 T^{2} - 46305372851786160 T^{3} + 118094246043 p^{7} T^{4} + 481320 p^{14} T^{5} + p^{21} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 221001 T + 146537503188 T^{2} - 117318812350687423 T^{3} + 146537503188 p^{7} T^{4} + 221001 p^{14} T^{5} + p^{21} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 1801254 T + 2334313879161 T^{2} - 1828515309774000636 T^{3} + 2334313879161 p^{7} T^{4} - 1801254 p^{14} T^{5} + p^{21} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 390828 T + 1410592098039 T^{2} - 851004153997014648 T^{3} + 1410592098039 p^{7} T^{4} - 390828 p^{14} T^{5} + p^{21} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 1990758 T + 1156997995845 T^{2} + 2940201149740237140 T^{3} + 1156997995845 p^{7} T^{4} - 1990758 p^{14} T^{5} + p^{21} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 107721 T + 9330454343610 T^{2} - 685745826060927125 T^{3} + 9330454343610 p^{7} T^{4} - 107721 p^{14} T^{5} + p^{21} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 3714339 T + 18982978466676 T^{2} - 39779799843926031091 T^{3} + 18982978466676 p^{7} T^{4} - 3714339 p^{14} T^{5} + p^{21} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 3292200 T + 21969621848373 T^{2} - 47963153031543572400 T^{3} + 21969621848373 p^{7} T^{4} - 3292200 p^{14} T^{5} + p^{21} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 583962 T + 1426589758839 T^{2} - 67853828372725252492 T^{3} + 1426589758839 p^{7} T^{4} - 583962 p^{14} T^{5} + p^{21} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 1488576 T + 48024849251469 T^{2} - 62780758073934985856 T^{3} + 48024849251469 p^{7} T^{4} - 1488576 p^{14} T^{5} + p^{21} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 11958726 T + 122211004925373 T^{2} + \)\(68\!\cdots\!92\)\( T^{3} + 122211004925373 p^{7} T^{4} + 11958726 p^{14} T^{5} + p^{21} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 564480 T + 122938538350587 T^{2} + 36166617283298019840 T^{3} + 122938538350587 p^{7} T^{4} + 564480 p^{14} T^{5} + p^{21} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 8308557 T + 186717656318022 T^{2} - \)\(87\!\cdots\!41\)\( T^{3} + 186717656318022 p^{7} T^{4} - 8308557 p^{14} T^{5} + p^{21} T^{6} \) |
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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
−9.323561665126322983596318027091, −8.890985069529165998983815323077, −8.578911853359564828297455287570, −8.452251218551034794787052255303, −7.87962010822926049522632266737, −7.85223517913373239002201954671, −7.46659143816586622491983101490, −6.83742405591916982976147692949, −6.78130070042583714404555831546, −6.67897527487871438430299403841, −5.64378606945454018887428389007, −5.57396297869986264666427814571, −5.42411046897937359993535691456, −4.39700931726138909317527301347, −4.38709883906854064684653405088, −4.22281915224765162409211665291, −3.38334996073198750817153754377, −3.29126618792892060418968051515, −3.11798351730673419375808310578, −2.27329984054672728420851613535, −2.17762608584706220069344512215, −1.74919835935072405898222806286, −1.18288007224565277230140157585, −0.71079289107837677449143503591, −0.67970248247781915200416970012,
0.67970248247781915200416970012, 0.71079289107837677449143503591, 1.18288007224565277230140157585, 1.74919835935072405898222806286, 2.17762608584706220069344512215, 2.27329984054672728420851613535, 3.11798351730673419375808310578, 3.29126618792892060418968051515, 3.38334996073198750817153754377, 4.22281915224765162409211665291, 4.38709883906854064684653405088, 4.39700931726138909317527301347, 5.42411046897937359993535691456, 5.57396297869986264666427814571, 5.64378606945454018887428389007, 6.67897527487871438430299403841, 6.78130070042583714404555831546, 6.83742405591916982976147692949, 7.46659143816586622491983101490, 7.85223517913373239002201954671, 7.87962010822926049522632266737, 8.452251218551034794787052255303, 8.578911853359564828297455287570, 8.890985069529165998983815323077, 9.323561665126322983596318027091
