| L(s) = 1 | + 5·3-s − 2·7-s − 19·9-s − 31·11-s + 8·13-s − 89·17-s − 87·19-s − 10·21-s − 122·23-s − 134·27-s + 84·29-s − 294·31-s − 155·33-s − 94·37-s + 40·39-s − 345·41-s − 412·43-s + 824·47-s − 333·49-s − 445·51-s + 74·53-s − 435·57-s − 1.04e3·59-s + 482·61-s + 38·63-s + 735·67-s − 610·69-s + ⋯ |
| L(s) = 1 | + 0.962·3-s − 0.107·7-s − 0.703·9-s − 0.849·11-s + 0.170·13-s − 1.26·17-s − 1.05·19-s − 0.103·21-s − 1.10·23-s − 0.955·27-s + 0.537·29-s − 1.70·31-s − 0.817·33-s − 0.417·37-s + 0.164·39-s − 1.31·41-s − 1.46·43-s + 2.55·47-s − 0.970·49-s − 1.22·51-s + 0.191·53-s − 1.01·57-s − 2.29·59-s + 1.01·61-s + 0.0759·63-s + 1.34·67-s − 1.06·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $S_4\times C_2$ | \( 1 - 5 T + 44 T^{2} - 181 T^{3} + 44 p^{3} T^{4} - 5 p^{6} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 2 T + 337 T^{2} - 4364 T^{3} + 337 p^{3} T^{4} + 2 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 31 T + 3508 T^{2} + 74047 T^{3} + 3508 p^{3} T^{4} + 31 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 8 T + 591 T^{2} + 156848 T^{3} + 591 p^{3} T^{4} - 8 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 89 T + 8734 T^{2} + 383789 T^{3} + 8734 p^{3} T^{4} + 89 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 87 T + 17172 T^{2} + 1201391 T^{3} + 17172 p^{3} T^{4} + 87 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 122 T + 37857 T^{2} + 2899940 T^{3} + 37857 p^{3} T^{4} + 122 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 84 T + 42447 T^{2} - 3818824 T^{3} + 42447 p^{3} T^{4} - 84 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 294 T + 89193 T^{2} + 15888508 T^{3} + 89193 p^{3} T^{4} + 294 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 94 T + 25619 T^{2} - 974636 T^{3} + 25619 p^{3} T^{4} + 94 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 345 T + 173478 T^{2} + 33313165 T^{3} + 173478 p^{3} T^{4} + 345 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 412 T + 230201 T^{2} + 56312872 T^{3} + 230201 p^{3} T^{4} + 412 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 824 T + 454493 T^{2} - 159690000 T^{3} + 454493 p^{3} T^{4} - 824 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 74 T + 242291 T^{2} - 22110396 T^{3} + 242291 p^{3} T^{4} - 74 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 1040 T + 622137 T^{2} + 287028320 T^{3} + 622137 p^{3} T^{4} + 1040 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 482 T + 516203 T^{2} - 138176204 T^{3} + 516203 p^{3} T^{4} - 482 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 735 T + 580956 T^{2} - 201276543 T^{3} + 580956 p^{3} T^{4} - 735 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 2048 T + 2303733 T^{2} + 1686995456 T^{3} + 2303733 p^{3} T^{4} + 2048 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 15 T + 940326 T^{2} - 7175365 T^{3} + 940326 p^{3} T^{4} + 15 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 998 T + 874777 T^{2} + 622163644 T^{3} + 874777 p^{3} T^{4} + 998 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 1221 T + 733620 T^{2} + 234081389 T^{3} + 733620 p^{3} T^{4} + 1221 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 1189 T + 1665246 T^{2} - 1034445553 T^{3} + 1665246 p^{3} T^{4} - 1189 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 1010 T - 724081 T^{2} - 1968025540 T^{3} - 724081 p^{3} T^{4} + 1010 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−9.183674338187058574457456730577, −8.643770745448045559187038676242, −8.631518297603473033381955391324, −8.596437350980618096506513542020, −8.055933535280615297273852410152, −7.73587747223916358042491649709, −7.67092005667711309083673738166, −7.13374053433191242860551439306, −6.84134109831190201734025737671, −6.62492099594118526435251016221, −6.22832334629825166585796571447, −5.84833862466170157586151412670, −5.71653360120193282112563158964, −5.24338579917369203923658798077, −4.93921049547134749750643312776, −4.63674042446308549332740454176, −4.02551760694343899778284952930, −3.95364981422226000725373511898, −3.62763939495586224810330775159, −2.88671134321781128301473437923, −2.85220033949001492452604871043, −2.56889216701834496224639773073, −1.93976828274267102165720809959, −1.77493289132216025333984762781, −1.22029348693172639227938317547, 0, 0, 0,
1.22029348693172639227938317547, 1.77493289132216025333984762781, 1.93976828274267102165720809959, 2.56889216701834496224639773073, 2.85220033949001492452604871043, 2.88671134321781128301473437923, 3.62763939495586224810330775159, 3.95364981422226000725373511898, 4.02551760694343899778284952930, 4.63674042446308549332740454176, 4.93921049547134749750643312776, 5.24338579917369203923658798077, 5.71653360120193282112563158964, 5.84833862466170157586151412670, 6.22832334629825166585796571447, 6.62492099594118526435251016221, 6.84134109831190201734025737671, 7.13374053433191242860551439306, 7.67092005667711309083673738166, 7.73587747223916358042491649709, 8.055933535280615297273852410152, 8.596437350980618096506513542020, 8.631518297603473033381955391324, 8.643770745448045559187038676242, 9.183674338187058574457456730577
