| L(s) = 1 | + 2-s + 5-s − 8-s + 10-s − 11-s + 13-s − 16-s + 2·17-s − 22-s − 23-s + 26-s − 29-s + 31-s + 2·34-s + 4·37-s − 40-s + 43-s − 46-s − 47-s − 49-s − 55-s − 58-s + 2·59-s + 62-s + 64-s + 65-s − 2·67-s + ⋯ |
| L(s) = 1 | + 2-s + 5-s − 8-s + 10-s − 11-s + 13-s − 16-s + 2·17-s − 22-s − 23-s + 26-s − 29-s + 31-s + 2·34-s + 4·37-s − 40-s + 43-s − 46-s − 47-s − 49-s − 55-s − 58-s + 2·59-s + 62-s + 64-s + 65-s − 2·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.586226794\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.586226794\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 37 | $C_1$ | \( ( 1 - T )^{4} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−9.164225326503075093526290437806, −8.526899519914217202915702190397, −8.111182062504364693057458488124, −8.023261178583268220212120078759, −7.61291398200573893463308492194, −7.16641169994682186097471998464, −6.40636365550386355958586814597, −6.21464575415869420849156469051, −5.83593196500109075263555327919, −5.74434651238776784845013659111, −5.30114540996283608005001692517, −4.85468627837880322833774618513, −4.34118311017043381265339896819, −4.04776543749108176063140305406, −3.50480834060103045059889593608, −3.12465080369670697783704529537, −2.55051594288242106292969631069, −2.34783654833587842096086494410, −1.46727638613897874483818129171, −0.892492290913744420472129927478,
0.892492290913744420472129927478, 1.46727638613897874483818129171, 2.34783654833587842096086494410, 2.55051594288242106292969631069, 3.12465080369670697783704529537, 3.50480834060103045059889593608, 4.04776543749108176063140305406, 4.34118311017043381265339896819, 4.85468627837880322833774618513, 5.30114540996283608005001692517, 5.74434651238776784845013659111, 5.83593196500109075263555327919, 6.21464575415869420849156469051, 6.40636365550386355958586814597, 7.16641169994682186097471998464, 7.61291398200573893463308492194, 8.023261178583268220212120078759, 8.111182062504364693057458488124, 8.526899519914217202915702190397, 9.164225326503075093526290437806
