| L(s) = 1 | − 6·2-s − 86·3-s − 804·4-s − 2.23e3·5-s + 516·6-s − 4.80e3·7-s + 6.79e3·8-s − 1.04e4·9-s + 1.34e4·10-s + 3.53e4·11-s + 6.91e4·12-s − 2.65e4·13-s + 2.88e4·14-s + 1.92e5·15-s + 3.90e5·16-s − 4.63e5·17-s + 6.27e4·18-s − 9.25e5·19-s + 1.79e6·20-s + 4.12e5·21-s − 2.11e5·22-s + 7.78e5·23-s − 5.84e5·24-s + 1.59e6·25-s + 1.59e5·26-s + 7.43e5·27-s + 3.86e6·28-s + ⋯ |
| L(s) = 1 | − 0.265·2-s − 0.612·3-s − 1.57·4-s − 1.60·5-s + 0.162·6-s − 0.755·7-s + 0.586·8-s − 0.531·9-s + 0.424·10-s + 0.727·11-s + 0.962·12-s − 0.257·13-s + 0.200·14-s + 0.981·15-s + 1.49·16-s − 1.34·17-s + 0.140·18-s − 1.62·19-s + 2.51·20-s + 0.463·21-s − 0.192·22-s + 0.579·23-s − 0.359·24-s + 0.815·25-s + 0.0683·26-s + 0.269·27-s + 1.18·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 7 | $C_1$ | \( ( 1 + p^{4} T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + 3 p T + 105 p^{3} T^{2} + 3 p^{10} T^{3} + p^{18} T^{4} \) |
| 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
−19.40675896893413344248306842164, −19.40423647552335104573695202448, −18.50866241667015281535077673490, −17.65173115358181152264424133471, −16.91114630353643452526918174418, −16.45240372921716870509386384917, −15.08781012950683421113327219960, −14.76812558417555503843247079453, −13.22126393463707655086164512156, −12.91204032733935345221037255583, −11.63042121779921486607189127837, −11.05889941982065285218625329280, −9.554205566700972309178900960496, −8.815004289195965606401040694219, −7.85307835004733678158176389773, −6.33630525032808966517598652196, −4.61918990308597522928980298957, −3.79837678420596625160071144778, 0, 0,
3.79837678420596625160071144778, 4.61918990308597522928980298957, 6.33630525032808966517598652196, 7.85307835004733678158176389773, 8.815004289195965606401040694219, 9.554205566700972309178900960496, 11.05889941982065285218625329280, 11.63042121779921486607189127837, 12.91204032733935345221037255583, 13.22126393463707655086164512156, 14.76812558417555503843247079453, 15.08781012950683421113327219960, 16.45240372921716870509386384917, 16.91114630353643452526918174418, 17.65173115358181152264424133471, 18.50866241667015281535077673490, 19.40423647552335104573695202448, 19.40675896893413344248306842164
