L(s) = 1 | + 3-s − 4-s − 11-s − 12-s + 2·23-s − 27-s + 31-s − 33-s + 2·37-s + 44-s − 47-s − 49-s + 2·53-s + 59-s + 64-s − 67-s + 2·69-s − 2·71-s − 81-s + 4·89-s − 2·92-s + 93-s − 97-s − 103-s + 108-s + 2·111-s − 113-s + ⋯ |
L(s) = 1 | + 3-s − 4-s − 11-s − 12-s + 2·23-s − 27-s + 31-s − 33-s + 2·37-s + 44-s − 47-s − 49-s + 2·53-s + 59-s + 64-s − 67-s + 2·69-s − 2·71-s − 81-s + 4·89-s − 2·92-s + 93-s − 97-s − 103-s + 108-s + 2·111-s − 113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.201706605\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.201706605\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_2$ | \( 1 + T + T^{2} \) |
good | 2 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_1$ | \( ( 1 - T )^{4} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
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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
−9.410985929848712615106790997803, −8.758412589084946704168195073635, −8.701952470505225732289211597338, −8.223371281512659741714185689608, −7.88052780456012402018296595116, −7.59522228250637489045989516451, −7.07865715415286509771607565017, −6.77191794842081015367782634723, −6.01732056864407023036729448778, −5.94360258333511489418202447152, −5.13858760138922981260008018134, −4.92364133138918527906150437846, −4.68314943332594180782099277072, −3.96925233145204346311140761045, −3.73230311150702403781378349087, −3.00898745401341516618974520873, −2.70152505316010835054888682479, −2.43856473416329474941547616601, −1.52901858271369773888075768574, −0.72137328655715355456477280561,
0.72137328655715355456477280561, 1.52901858271369773888075768574, 2.43856473416329474941547616601, 2.70152505316010835054888682479, 3.00898745401341516618974520873, 3.73230311150702403781378349087, 3.96925233145204346311140761045, 4.68314943332594180782099277072, 4.92364133138918527906150437846, 5.13858760138922981260008018134, 5.94360258333511489418202447152, 6.01732056864407023036729448778, 6.77191794842081015367782634723, 7.07865715415286509771607565017, 7.59522228250637489045989516451, 7.88052780456012402018296595116, 8.223371281512659741714185689608, 8.701952470505225732289211597338, 8.758412589084946704168195073635, 9.410985929848712615106790997803