| L(s) = 1 | − 2-s − 3-s + 5-s + 6-s + 7-s + 8-s + 9-s − 10-s − 11-s − 14-s − 15-s − 16-s − 18-s − 19-s − 21-s + 22-s − 24-s + 25-s − 2·27-s + 30-s + 33-s + 35-s − 2·37-s + 38-s + 40-s + 2·41-s + 42-s + ⋯ |
| L(s) = 1 | − 2-s − 3-s + 5-s + 6-s + 7-s + 8-s + 9-s − 10-s − 11-s − 14-s − 15-s − 16-s − 18-s − 19-s − 21-s + 22-s − 24-s + 25-s − 2·27-s + 30-s + 33-s + 35-s − 2·37-s + 38-s + 40-s + 2·41-s + 42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1317904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1317904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4878348991\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4878348991\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 - T + T^{2} \) |
| 41 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−10.37321964676910319257055147918, −9.746447742330723522468512071800, −9.538139982747020315127533888076, −8.969755346835513017661856541051, −8.662417026945543986841090108095, −8.160775831827835085407142989910, −7.85290978032906959231405822599, −7.29045281181377241814306036779, −7.03541724368761814260918152495, −6.54556642574320883872358064586, −5.75550773140910433551476629299, −5.70085743290770348017164171451, −5.06437340421326642834258678823, −4.89034434730231048082670550569, −4.11489247448533858616863579555, −3.91173057400785838861138862363, −2.69498805889845761080913673024, −2.07001008651529878065695396188, −1.73217175561221329925690682096, −0.827225766858395063243323841899,
0.827225766858395063243323841899, 1.73217175561221329925690682096, 2.07001008651529878065695396188, 2.69498805889845761080913673024, 3.91173057400785838861138862363, 4.11489247448533858616863579555, 4.89034434730231048082670550569, 5.06437340421326642834258678823, 5.70085743290770348017164171451, 5.75550773140910433551476629299, 6.54556642574320883872358064586, 7.03541724368761814260918152495, 7.29045281181377241814306036779, 7.85290978032906959231405822599, 8.160775831827835085407142989910, 8.662417026945543986841090108095, 8.969755346835513017661856541051, 9.538139982747020315127533888076, 9.746447742330723522468512071800, 10.37321964676910319257055147918
