| L(s) = 1 | − 4-s + 2·7-s − 2·19-s + 25-s − 2·28-s − 2·43-s + 49-s + 2·61-s + 64-s + 4·73-s + 2·76-s − 100-s + 121-s + 127-s + 131-s − 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 2·172-s + 173-s + 2·175-s + ⋯ |
| L(s) = 1 | − 4-s + 2·7-s − 2·19-s + 25-s − 2·28-s − 2·43-s + 49-s + 2·61-s + 64-s + 4·73-s + 2·76-s − 100-s + 121-s + 127-s + 131-s − 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 2·172-s + 173-s + 2·175-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368521 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368521 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.014906846\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.014906846\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$ | \( ( 1 - T )^{4} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 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
−9.650650723555546499837889486937, −9.591868969544966823045710407033, −8.730148848928685601726001051578, −8.558946423750725839661811112207, −8.474636817366710069209099881886, −7.991714930447270107550244874221, −7.68887658138772391712740438446, −6.90991640501894632064400396441, −6.69439800160805780778358107061, −6.28457714275770450267394569604, −5.53377714286828253659788043297, −4.99039715913136716743313058003, −4.98429692524604869927217162874, −4.55171657735019891948331621602, −3.98089474169719410148789408788, −3.69363420196059329012473283860, −2.83438931511686711694233439477, −1.95041990697815776784972040938, −1.94961909288261607304824594752, −0.854543512450298525288623978530,
0.854543512450298525288623978530, 1.94961909288261607304824594752, 1.95041990697815776784972040938, 2.83438931511686711694233439477, 3.69363420196059329012473283860, 3.98089474169719410148789408788, 4.55171657735019891948331621602, 4.98429692524604869927217162874, 4.99039715913136716743313058003, 5.53377714286828253659788043297, 6.28457714275770450267394569604, 6.69439800160805780778358107061, 6.90991640501894632064400396441, 7.68887658138772391712740438446, 7.991714930447270107550244874221, 8.474636817366710069209099881886, 8.558946423750725839661811112207, 8.730148848928685601726001051578, 9.591868969544966823045710407033, 9.650650723555546499837889486937
