L(s) = 1 | − 2·3-s − 3·4-s + 9-s + 6·12-s − 10·13-s + 5·16-s + 6·25-s + 4·27-s − 3·36-s + 6·37-s + 20·39-s − 10·48-s + 5·49-s + 30·52-s − 3·64-s + 8·67-s − 12·75-s − 11·81-s − 18·100-s − 12·108-s + 6·109-s − 12·111-s − 10·117-s + 14·121-s + 127-s + 131-s + 137-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 3/2·4-s + 1/3·9-s + 1.73·12-s − 2.77·13-s + 5/4·16-s + 6/5·25-s + 0.769·27-s − 1/2·36-s + 0.986·37-s + 3.20·39-s − 1.44·48-s + 5/7·49-s + 4.16·52-s − 3/8·64-s + 0.977·67-s − 1.38·75-s − 1.22·81-s − 9/5·100-s − 1.15·108-s + 0.574·109-s − 1.13·111-s − 0.924·117-s + 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 221841 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221841 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 157 | $C_2$ | \( 1 + 22 T + p T^{2} \) |
good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 158 T^{2} + p^{2} T^{4} \) |
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
−8.864881711317206882972246910233, −8.400417992219280667154508371426, −7.81827102714720521501936312242, −7.29173528114820563218340273940, −6.93593381707597865366646633277, −6.28122666685972414921774145186, −5.65337943506841464672887525807, −5.17382072459828148996456813455, −4.80809444586124065077345108731, −4.58827031874147663510896677958, −3.87111710558908404869292116695, −2.92807888429295739114534520463, −2.33013519377393677733042476987, −0.861343348717887878779047412541, 0,
0.861343348717887878779047412541, 2.33013519377393677733042476987, 2.92807888429295739114534520463, 3.87111710558908404869292116695, 4.58827031874147663510896677958, 4.80809444586124065077345108731, 5.17382072459828148996456813455, 5.65337943506841464672887525807, 6.28122666685972414921774145186, 6.93593381707597865366646633277, 7.29173528114820563218340273940, 7.81827102714720521501936312242, 8.400417992219280667154508371426, 8.864881711317206882972246910233