| L(s) = 1 | − 10·5-s + 14·7-s + 52·13-s − 26·17-s + 28·19-s − 164·23-s − 111·25-s − 174·29-s + 318·31-s − 140·35-s + 296·37-s + 118·41-s − 260·43-s − 204·47-s − 253·49-s − 1.08e3·53-s − 196·59-s + 1.53e3·61-s − 520·65-s − 660·67-s + 852·71-s − 478·73-s + 22·79-s − 1.13e3·83-s + 260·85-s − 110·89-s + 728·91-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.755·7-s + 1.10·13-s − 0.370·17-s + 0.338·19-s − 1.48·23-s − 0.887·25-s − 1.11·29-s + 1.84·31-s − 0.676·35-s + 1.31·37-s + 0.449·41-s − 0.922·43-s − 0.633·47-s − 0.737·49-s − 2.81·53-s − 0.432·59-s + 3.22·61-s − 0.992·65-s − 1.20·67-s + 1.42·71-s − 0.766·73-s + 0.0313·79-s − 1.50·83-s + 0.331·85-s − 0.131·89-s + 0.838·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7148942318\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7148942318\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $S_4\times C_2$ | \( 1 + 2 p T + 211 T^{2} + 2396 T^{3} + 211 p^{3} T^{4} + 2 p^{7} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 2 p T + 449 T^{2} - 3788 T^{3} + 449 p^{3} T^{4} - 2 p^{7} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 107 p T^{2} - 49152 T^{3} + 107 p^{4} T^{4} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 4 p T + 5487 T^{2} - 172616 T^{3} + 5487 p^{3} T^{4} - 4 p^{7} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 26 T + 3615 T^{2} - 222100 T^{3} + 3615 p^{3} T^{4} + 26 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 28 T + 9489 T^{2} - 658856 T^{3} + 9489 p^{3} T^{4} - 28 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 164 T + 42885 T^{2} + 4036280 T^{3} + 42885 p^{3} T^{4} + 164 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 6 p T + 77131 T^{2} + 8426004 T^{3} + 77131 p^{3} T^{4} + 6 p^{7} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 318 T + 93849 T^{2} - 15197452 T^{3} + 93849 p^{3} T^{4} - 318 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 8 p T + 105879 T^{2} - 29903632 T^{3} + 105879 p^{3} T^{4} - 8 p^{7} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 118 T + 89463 T^{2} + 3720620 T^{3} + 89463 p^{3} T^{4} - 118 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 260 T + 157545 T^{2} + 32742232 T^{3} + 157545 p^{3} T^{4} + 260 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 204 T + 283677 T^{2} + 40395048 T^{3} + 283677 p^{3} T^{4} + 204 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 1086 T + 736099 T^{2} + 344026068 T^{3} + 736099 p^{3} T^{4} + 1086 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 196 T + 566137 T^{2} + 71984984 T^{3} + 566137 p^{3} T^{4} + 196 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 1536 T + 1409583 T^{2} - 802126848 T^{3} + 1409583 p^{3} T^{4} - 1536 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 660 T + 592833 T^{2} + 364778232 T^{3} + 592833 p^{3} T^{4} + 660 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 12 p T + 1006773 T^{2} - 524795352 T^{3} + 1006773 p^{3} T^{4} - 12 p^{7} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 478 T + 911095 T^{2} + 251066948 T^{3} + 911095 p^{3} T^{4} + 478 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 22 T + 1407593 T^{2} - 13791100 T^{3} + 1407593 p^{3} T^{4} - 22 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 1136 T + 2100385 T^{2} + 1337053600 T^{3} + 2100385 p^{3} T^{4} + 1136 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 110 T + 2073543 T^{2} + 156516836 T^{3} + 2073543 p^{3} T^{4} + 110 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 1222 T + 2989679 T^{2} + 2155770388 T^{3} + 2989679 p^{3} T^{4} + 1222 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−7.945066488648513771639825870649, −7.31804025920619724697011036400, −7.29291578644939668147593265495, −6.82831650706071106877251428568, −6.51967495582573731412024093307, −6.32143191514298020264731830913, −6.09791993292967328864996146751, −5.88915394973023718031189601268, −5.41765775799692119809725133327, −5.24844790082389519866567378358, −4.95143584382472748902946692083, −4.58144922842678958115620180271, −4.28850715255389913340941914681, −3.92005224104757685305385291567, −3.91252647560876906292534929395, −3.74580905898083027413316148088, −2.93937194439963552182252509358, −2.87558580978803821072781452774, −2.73650201248061846516988857273, −1.87086694723699650879598079078, −1.72792993172750274937496382467, −1.63165942303202124481769589764, −0.997978014035165967829543820555, −0.62065278133666110248169914772, −0.13092154094516947552086111889,
0.13092154094516947552086111889, 0.62065278133666110248169914772, 0.997978014035165967829543820555, 1.63165942303202124481769589764, 1.72792993172750274937496382467, 1.87086694723699650879598079078, 2.73650201248061846516988857273, 2.87558580978803821072781452774, 2.93937194439963552182252509358, 3.74580905898083027413316148088, 3.91252647560876906292534929395, 3.92005224104757685305385291567, 4.28850715255389913340941914681, 4.58144922842678958115620180271, 4.95143584382472748902946692083, 5.24844790082389519866567378358, 5.41765775799692119809725133327, 5.88915394973023718031189601268, 6.09791993292967328864996146751, 6.32143191514298020264731830913, 6.51967495582573731412024093307, 6.82831650706071106877251428568, 7.29291578644939668147593265495, 7.31804025920619724697011036400, 7.945066488648513771639825870649
