| L(s) = 1 | + 2·7-s + 7·13-s − 8·19-s + 5·25-s − 22·31-s − 2·37-s + 5·43-s − 11·49-s + 13·61-s + 5·67-s + 7·73-s − 13·79-s + 14·91-s − 14·97-s + 26·103-s − 2·109-s − 22·121-s + 127-s + 131-s − 16·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 1.94·13-s − 1.83·19-s + 25-s − 3.95·31-s − 0.328·37-s + 0.762·43-s − 1.57·49-s + 1.66·61-s + 0.610·67-s + 0.819·73-s − 1.46·79-s + 1.46·91-s − 1.42·97-s + 2.56·103-s − 0.191·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s − 1.38·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.210440835\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.210440835\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.909651498756595813186719366062, −8.481380592290244429273515498935, −8.443902891164979404425085391891, −8.067176148519715323875600612147, −7.40441181731652690071841014593, −7.14050091451346010672859184291, −6.79812311692226590575953247065, −6.25126405539364527328459067714, −5.99249343042853713846521599854, −5.55845607058151059496893826361, −5.14904996723655646782918689430, −4.75148266802816772793363610123, −4.15856331671155656265568194913, −3.75754052939657342197403457249, −3.61025810929865884953444879209, −2.93321309807980855637419691174, −2.18232044308614782165028871657, −1.78167645699854059480313143426, −1.40860371051596851626892960750, −0.48344629813890547028717724100,
0.48344629813890547028717724100, 1.40860371051596851626892960750, 1.78167645699854059480313143426, 2.18232044308614782165028871657, 2.93321309807980855637419691174, 3.61025810929865884953444879209, 3.75754052939657342197403457249, 4.15856331671155656265568194913, 4.75148266802816772793363610123, 5.14904996723655646782918689430, 5.55845607058151059496893826361, 5.99249343042853713846521599854, 6.25126405539364527328459067714, 6.79812311692226590575953247065, 7.14050091451346010672859184291, 7.40441181731652690071841014593, 8.067176148519715323875600612147, 8.443902891164979404425085391891, 8.481380592290244429273515498935, 8.909651498756595813186719366062
