| L(s) = 1 | + 3-s + 5-s + 7-s + 15-s + 21-s + 23-s − 27-s − 29-s + 35-s + 41-s + 2·43-s + 47-s + 49-s − 61-s − 67-s + 69-s − 81-s − 83-s − 87-s − 2·89-s + 2·101-s − 2·103-s + 105-s + 2·107-s + 2·109-s + 115-s − 121-s + ⋯ |
| L(s) = 1 | + 3-s + 5-s + 7-s + 15-s + 21-s + 23-s − 27-s − 29-s + 35-s + 41-s + 2·43-s + 47-s + 49-s − 61-s − 67-s + 69-s − 81-s − 83-s − 87-s − 2·89-s + 2·101-s − 2·103-s + 105-s + 2·107-s + 2·109-s + 115-s − 121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.709331731\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.709331731\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 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
−9.223826718305855265114034195060, −8.911404599159117196362578289378, −8.480369050468391839810714468529, −8.052520986540843387610259395733, −7.51327550696187853948440466282, −7.42534117913834486056216984344, −7.18621530404378794048179723054, −6.35293503791239846851923782925, −5.92320148961871584019092971743, −5.86720076966175889530050794394, −5.24341693315974065382397879082, −5.00557815200628854761433989997, −4.27769218216159212974723174349, −4.09654127510504351517244595889, −3.55658777993906907654344569534, −2.74022108219464296623192514342, −2.73326392370793457871694347760, −2.10711834991941699631872380659, −1.64509517683527146491139362783, −1.05835520682313524296989683089,
1.05835520682313524296989683089, 1.64509517683527146491139362783, 2.10711834991941699631872380659, 2.73326392370793457871694347760, 2.74022108219464296623192514342, 3.55658777993906907654344569534, 4.09654127510504351517244595889, 4.27769218216159212974723174349, 5.00557815200628854761433989997, 5.24341693315974065382397879082, 5.86720076966175889530050794394, 5.92320148961871584019092971743, 6.35293503791239846851923782925, 7.18621530404378794048179723054, 7.42534117913834486056216984344, 7.51327550696187853948440466282, 8.052520986540843387610259395733, 8.480369050468391839810714468529, 8.911404599159117196362578289378, 9.223826718305855265114034195060
