| L(s) = 1 | + 6·2-s − 804·4-s + 2.23e3·5-s − 4.80e3·7-s − 6.79e3·8-s + 1.34e4·10-s − 3.53e4·11-s − 2.65e4·13-s − 2.88e4·14-s + 3.90e5·16-s + 4.63e5·17-s − 9.25e5·19-s − 1.79e6·20-s − 2.11e5·22-s − 7.78e5·23-s + 1.59e6·25-s − 1.59e5·26-s + 3.86e6·28-s + 1.00e7·29-s + 2.46e6·31-s + 4.00e6·32-s + 2.78e6·34-s − 1.07e7·35-s + 3.07e7·37-s − 5.55e6·38-s − 1.52e7·40-s + 1.91e7·41-s + ⋯ |
| L(s) = 1 | + 0.265·2-s − 1.57·4-s + 1.60·5-s − 0.755·7-s − 0.586·8-s + 0.424·10-s − 0.727·11-s − 0.257·13-s − 0.200·14-s + 1.49·16-s + 1.34·17-s − 1.62·19-s − 2.51·20-s − 0.192·22-s − 0.579·23-s + 0.815·25-s − 0.0683·26-s + 1.18·28-s + 2.62·29-s + 0.479·31-s + 0.675·32-s + 0.357·34-s − 1.21·35-s + 2.69·37-s − 0.431·38-s − 0.938·40-s + 1.05·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.200458946\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.200458946\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p^{4} T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - 3 p T + 105 p^{3} T^{2} - 3 p^{10} T^{3} + p^{18} T^{4} \) |
| 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
−13.30395682107059227481721473065, −13.02330884061677840197895484485, −12.38763670566049960785368817548, −12.07258822678794881184054119853, −10.68329679775750014404255691734, −10.27478794030548957643187315349, −9.811751206294540680825023215048, −9.515105265928074291162687888730, −8.723354408451158818185602100528, −8.293268739667757989138747577094, −7.45648376424714032787229267114, −6.34317591265805944725595352958, −5.83420409501913154825657017499, −5.55020489777273626156891487216, −4.32609879453086827627036105595, −4.32113333999697713709168690980, −2.79461937704234022375712843980, −2.53879406179636735069622925640, −1.14536603437960030537564022318, −0.50826650036454908993081422119,
0.50826650036454908993081422119, 1.14536603437960030537564022318, 2.53879406179636735069622925640, 2.79461937704234022375712843980, 4.32113333999697713709168690980, 4.32609879453086827627036105595, 5.55020489777273626156891487216, 5.83420409501913154825657017499, 6.34317591265805944725595352958, 7.45648376424714032787229267114, 8.293268739667757989138747577094, 8.723354408451158818185602100528, 9.515105265928074291162687888730, 9.811751206294540680825023215048, 10.27478794030548957643187315349, 10.68329679775750014404255691734, 12.07258822678794881184054119853, 12.38763670566049960785368817548, 13.02330884061677840197895484485, 13.30395682107059227481721473065
