| L(s) = 1 | − 8·3-s − 42·7-s + 32·9-s − 6·11-s + 100·13-s − 28·17-s − 120·19-s + 336·21-s − 69·23-s − 239·25-s − 148·27-s − 128·29-s + 76·31-s + 48·33-s − 212·37-s − 800·39-s − 580·41-s + 20·43-s + 396·47-s + 283·49-s + 224·51-s − 122·53-s + 960·57-s − 632·59-s + 338·61-s − 1.34e3·63-s − 442·67-s + ⋯ |
| L(s) = 1 | − 1.53·3-s − 2.26·7-s + 1.18·9-s − 0.164·11-s + 2.13·13-s − 0.399·17-s − 1.44·19-s + 3.49·21-s − 0.625·23-s − 1.91·25-s − 1.05·27-s − 0.819·29-s + 0.440·31-s + 0.253·33-s − 0.941·37-s − 3.28·39-s − 2.20·41-s + 0.0709·43-s + 1.22·47-s + 0.825·49-s + 0.615·51-s − 0.316·53-s + 2.23·57-s − 1.39·59-s + 0.709·61-s − 2.68·63-s − 0.805·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49836032 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49836032 ^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 + p T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 148 T^{3} + 32 p^{3} T^{4} + 8 p^{6} T^{5} + p^{9} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + 239 T^{2} - 552 T^{3} + 239 p^{3} T^{4} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 6 p T + 1481 T^{2} + 29100 T^{3} + 1481 p^{3} T^{4} + 6 p^{7} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 6 T + 357 T^{2} - 73452 T^{3} + 357 p^{3} T^{4} + 6 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 100 T + 8612 T^{2} - 415006 T^{3} + 8612 p^{3} T^{4} - 100 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 28 T + 11379 T^{2} + 218752 T^{3} + 11379 p^{3} T^{4} + 28 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 120 T + 22229 T^{2} + 1650144 T^{3} + 22229 p^{3} T^{4} + 120 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 128 T + 73620 T^{2} + 6012410 T^{3} + 73620 p^{3} T^{4} + 128 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 76 T + 74612 T^{2} - 4910440 T^{3} + 74612 p^{3} T^{4} - 76 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 212 T + 142519 T^{2} + 21541408 T^{3} + 142519 p^{3} T^{4} + 212 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 580 T + 286636 T^{2} + 79438678 T^{3} + 286636 p^{3} T^{4} + 580 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 20 T + 45049 T^{2} + 22781192 T^{3} + 45049 p^{3} T^{4} - 20 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 396 T + 322364 T^{2} - 76585512 T^{3} + 322364 p^{3} T^{4} - 396 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 122 T + 149491 T^{2} + 64710044 T^{3} + 149491 p^{3} T^{4} + 122 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 632 T + 682857 T^{2} + 250521872 T^{3} + 682857 p^{3} T^{4} + 632 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 338 T + 698459 T^{2} - 151418252 T^{3} + 698459 p^{3} T^{4} - 338 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 442 T + 295309 T^{2} + 159890060 T^{3} + 295309 p^{3} T^{4} + 442 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 888 T + 853800 T^{2} + 485139624 T^{3} + 853800 p^{3} T^{4} + 888 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 376 T + 123364 T^{2} + 139115710 T^{3} + 123364 p^{3} T^{4} - 376 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 1540 T + 2120833 T^{2} + 1585051256 T^{3} + 2120833 p^{3} T^{4} + 1540 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 454 T + 821989 T^{2} + 278132212 T^{3} + 821989 p^{3} T^{4} + 454 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 810 T + 1992291 T^{2} + 1141870932 T^{3} + 1992291 p^{3} T^{4} + 810 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 1284 T + 3062459 T^{2} - 2365948272 T^{3} + 3062459 p^{3} T^{4} - 1284 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−10.39604720322775766295359319279, −9.862966450048822897016985879442, −9.828794688491228103735667582246, −9.475440368834841384699049908329, −8.843492495602802182734592471347, −8.775145214828924821466178651761, −8.509926169355764417663879075416, −8.015004479475885786927040133414, −7.60169465235456851205949259403, −7.20041730132650478217695277671, −6.82491394083077506132171158599, −6.39051347444954895041936360515, −6.37552490331860313261488998610, −6.03945354137867676419767586012, −5.72839020709858406583199481614, −5.62204292135530454692552166400, −4.95941400768899784680602171956, −4.40723322299110760344222090382, −4.05483077715278300673123612770, −3.78806706814283077988685296143, −3.27142701363048157717744835606, −3.11307018615571208681029530005, −2.16998540975246566329355251052, −1.74291718620808992461598726180, −1.19029050569334906769832206404, 0, 0, 0,
1.19029050569334906769832206404, 1.74291718620808992461598726180, 2.16998540975246566329355251052, 3.11307018615571208681029530005, 3.27142701363048157717744835606, 3.78806706814283077988685296143, 4.05483077715278300673123612770, 4.40723322299110760344222090382, 4.95941400768899784680602171956, 5.62204292135530454692552166400, 5.72839020709858406583199481614, 6.03945354137867676419767586012, 6.37552490331860313261488998610, 6.39051347444954895041936360515, 6.82491394083077506132171158599, 7.20041730132650478217695277671, 7.60169465235456851205949259403, 8.015004479475885786927040133414, 8.509926169355764417663879075416, 8.775145214828924821466178651761, 8.843492495602802182734592471347, 9.475440368834841384699049908329, 9.828794688491228103735667582246, 9.862966450048822897016985879442, 10.39604720322775766295359319279
