| L(s) = 1 | + 2·3-s + 9-s + 2·13-s − 4·19-s − 2·25-s − 4·27-s + 2·31-s + 4·39-s − 20·43-s + 49-s − 8·57-s − 2·61-s − 2·67-s + 6·73-s − 4·75-s − 2·79-s − 11·81-s + 4·93-s − 12·97-s + 8·103-s + 14·109-s + 2·117-s − 10·121-s + 127-s − 40·129-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s + 0.554·13-s − 0.917·19-s − 2/5·25-s − 0.769·27-s + 0.359·31-s + 0.640·39-s − 3.04·43-s + 1/7·49-s − 1.05·57-s − 0.256·61-s − 0.244·67-s + 0.702·73-s − 0.461·75-s − 0.225·79-s − 1.22·81-s + 0.414·93-s − 1.21·97-s + 0.788·103-s + 1.34·109-s + 0.184·117-s − 0.909·121-s + 0.0887·127-s − 3.52·129-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1192464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1192464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 114 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 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
−8.043945394042314888922317801847, −7.37231131387349515731445760589, −7.09757869365676625006137667716, −6.44240832191410372311240555245, −6.18739967856341767734195924203, −5.65669662563406255614726924467, −4.97931365434473032081556124465, −4.66522715600328581707060439735, −3.90711236554140198013672238587, −3.61971784540853340917696518791, −3.12002412558388795839852221862, −2.50298959733082881194172826935, −1.96684834650446670595982820502, −1.34749550489351632879545747408, 0,
1.34749550489351632879545747408, 1.96684834650446670595982820502, 2.50298959733082881194172826935, 3.12002412558388795839852221862, 3.61971784540853340917696518791, 3.90711236554140198013672238587, 4.66522715600328581707060439735, 4.97931365434473032081556124465, 5.65669662563406255614726924467, 6.18739967856341767734195924203, 6.44240832191410372311240555245, 7.09757869365676625006137667716, 7.37231131387349515731445760589, 8.043945394042314888922317801847
