| L(s) = 1 | + 86·3-s − 2.23e3·5-s + 4.80e3·7-s − 1.04e4·9-s − 3.53e4·11-s − 2.65e4·13-s − 1.92e5·15-s − 4.63e5·17-s + 9.25e5·19-s + 4.12e5·21-s − 7.78e5·23-s + 1.59e6·25-s − 7.43e5·27-s − 1.00e7·29-s − 2.46e6·31-s − 3.03e6·33-s − 1.07e7·35-s + 3.07e7·37-s − 2.28e6·39-s − 1.91e7·41-s − 4.06e6·43-s + 2.34e7·45-s + 8.21e7·47-s + 1.72e7·49-s − 3.98e7·51-s − 5.51e7·53-s + 7.90e7·55-s + ⋯ |
| L(s) = 1 | + 0.612·3-s − 1.60·5-s + 0.755·7-s − 0.531·9-s − 0.727·11-s − 0.257·13-s − 0.981·15-s − 1.34·17-s + 1.62·19-s + 0.463·21-s − 0.579·23-s + 0.815·25-s − 0.269·27-s − 2.62·29-s − 0.479·31-s − 0.445·33-s − 1.21·35-s + 2.69·37-s − 0.157·39-s − 1.05·41-s − 0.181·43-s + 0.851·45-s + 2.45·47-s + 3/7·49-s − 0.825·51-s − 0.960·53-s + 1.16·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p^{4} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 86 T + 5954 p T^{2} - 86 p^{9} T^{3} + p^{18} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 2238 T + 3416586 T^{2} + 2238 p^{9} T^{3} + p^{18} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 35316 T + 2892681078 T^{2} + 35316 p^{9} T^{3} + p^{18} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 26530 T - 1541163822 T^{2} + 26530 p^{9} T^{3} + p^{18} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 463920 T + 273833245726 T^{2} + 463920 p^{9} T^{3} + p^{18} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 925426 T + 858791487510 T^{2} - 925426 p^{9} T^{3} + p^{18} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 778128 T + 3473691840430 T^{2} + 778128 p^{9} T^{3} + p^{18} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 10003584 T + 52302031706070 T^{2} + 10003584 p^{9} T^{3} + p^{18} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2467260 T + 49371575832542 T^{2} + 2467260 p^{9} T^{3} + p^{18} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 30735552 T + 484209298874630 T^{2} - 30735552 p^{9} T^{3} + p^{18} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 19103448 T + 602984827739166 T^{2} + 19103448 p^{9} T^{3} + p^{18} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4065100 T + 797231337676374 T^{2} + 4065100 p^{9} T^{3} + p^{18} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 82195020 T + 3721245520696702 T^{2} - 82195020 p^{9} T^{3} + p^{18} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 55189812 T + 3123778356606670 T^{2} + 55189812 p^{9} T^{3} + p^{18} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 7069218 T + 16866494212382134 T^{2} - 7069218 p^{9} T^{3} + p^{18} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 44316386 T + 21654336818123658 T^{2} - 44316386 p^{9} T^{3} + p^{18} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 241921336 T + 59516583718815510 T^{2} - 241921336 p^{9} T^{3} + p^{18} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 206493816 T + 58491352612128526 T^{2} + 206493816 p^{9} T^{3} + p^{18} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 499153188 T + 178474458263805254 T^{2} + 499153188 p^{9} T^{3} + p^{18} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 5930824 p T + 239633073722978334 T^{2} + 5930824 p^{10} T^{3} + p^{18} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 444023958 T + 333438010641681622 T^{2} + 444023958 p^{9} T^{3} + p^{18} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 636267396 T + 801539802340191990 T^{2} - 636267396 p^{9} T^{3} + p^{18} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1632716064 T + 2180562419544849758 T^{2} + 1632716064 p^{9} T^{3} + p^{18} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.55206976575284135103757290091, −11.26374988876998013529395826435, −10.75726542089281587034065612843, −9.941204736437172287083283808190, −9.205560221289822248806860569574, −8.933081711445210727584802973744, −8.036299946020655839169291753179, −7.910607877853971228976703146554, −7.44378078047853786658053056306, −6.92317963741079572054600530899, −5.63396343998845078454833283228, −5.51060645804642308078069050117, −4.30825304157260694289399628799, −4.21557918617837923510654951642, −3.34772700397629722650467593168, −2.70343406214420550822216851917, −2.07548256727593094130680212689, −1.11775391731785539198997811547, 0, 0,
1.11775391731785539198997811547, 2.07548256727593094130680212689, 2.70343406214420550822216851917, 3.34772700397629722650467593168, 4.21557918617837923510654951642, 4.30825304157260694289399628799, 5.51060645804642308078069050117, 5.63396343998845078454833283228, 6.92317963741079572054600530899, 7.44378078047853786658053056306, 7.910607877853971228976703146554, 8.036299946020655839169291753179, 8.933081711445210727584802973744, 9.205560221289822248806860569574, 9.941204736437172287083283808190, 10.75726542089281587034065612843, 11.26374988876998013529395826435, 11.55206976575284135103757290091
