| L(s) = 1 | − 4-s − 9-s − 2·11-s + 16-s − 8·19-s − 12·29-s + 36-s − 12·41-s + 2·44-s − 2·49-s − 24·59-s − 28·61-s − 64-s − 24·71-s + 8·76-s + 8·79-s + 81-s − 20·89-s + 2·99-s + 28·101-s + 12·109-s + 12·116-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1/3·9-s − 0.603·11-s + 1/4·16-s − 1.83·19-s − 2.22·29-s + 1/6·36-s − 1.87·41-s + 0.301·44-s − 2/7·49-s − 3.12·59-s − 3.58·61-s − 1/8·64-s − 2.84·71-s + 0.917·76-s + 0.900·79-s + 1/9·81-s − 2.11·89-s + 0.201·99-s + 2.78·101-s + 1.14·109-s + 1.11·116-s + 3/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2722500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2722500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 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
−8.981352038600481740689080715191, −8.980848877051300644645337099655, −8.385557687984348992794682633991, −8.058131967779387883020981639814, −7.54347591747245298412792612523, −7.36056275940929173931600693863, −6.78989845527356060266276113225, −6.08376500651138318940357171028, −6.01768084346551104433706628457, −5.64521412953945541145809675061, −4.76590854924204863110357672172, −4.72741729625368473557963500877, −4.28981941884877696066351771314, −3.48955354535357901536994797041, −3.27166121372302567731060941628, −2.65628972583205912832328670495, −1.78032040572762581040858560484, −1.65001841672879685493273212280, 0, 0,
1.65001841672879685493273212280, 1.78032040572762581040858560484, 2.65628972583205912832328670495, 3.27166121372302567731060941628, 3.48955354535357901536994797041, 4.28981941884877696066351771314, 4.72741729625368473557963500877, 4.76590854924204863110357672172, 5.64521412953945541145809675061, 6.01768084346551104433706628457, 6.08376500651138318940357171028, 6.78989845527356060266276113225, 7.36056275940929173931600693863, 7.54347591747245298412792612523, 8.058131967779387883020981639814, 8.385557687984348992794682633991, 8.980848877051300644645337099655, 8.981352038600481740689080715191
