| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s − 12-s + 16-s − 18-s − 6·19-s − 23-s + 24-s − 4·25-s − 27-s − 4·29-s − 32-s + 36-s + 6·38-s + 7·43-s + 46-s − 8·47-s − 48-s + 11·49-s + 4·50-s + 54-s + 6·57-s + 4·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.288·12-s + 1/4·16-s − 0.235·18-s − 1.37·19-s − 0.208·23-s + 0.204·24-s − 4/5·25-s − 0.192·27-s − 0.742·29-s − 0.176·32-s + 1/6·36-s + 0.973·38-s + 1.06·43-s + 0.147·46-s − 1.16·47-s − 0.144·48-s + 11/7·49-s + 0.565·50-s + 0.136·54-s + 0.794·57-s + 0.525·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 501120 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 501120 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + T + p T^{2} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 3 T + p T^{2} ) \) |
| good | 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 137 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 140 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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
−8.303153144335519193479190862263, −7.85861866528850393005294607551, −7.40221238766697726306191743187, −6.98614619309963994937005382629, −6.38631796325832338902523712010, −6.13823186998079257160847440568, −5.59588848667035618285338752564, −5.08346280753484696162467624003, −4.44784120840465394532543755177, −3.88755878571122145508008365230, −3.43883485187451816188100082712, −2.32750897177090168421868218004, −2.11489491300250239473846026111, −1.05727198181310832118400986849, 0,
1.05727198181310832118400986849, 2.11489491300250239473846026111, 2.32750897177090168421868218004, 3.43883485187451816188100082712, 3.88755878571122145508008365230, 4.44784120840465394532543755177, 5.08346280753484696162467624003, 5.59588848667035618285338752564, 6.13823186998079257160847440568, 6.38631796325832338902523712010, 6.98614619309963994937005382629, 7.40221238766697726306191743187, 7.85861866528850393005294607551, 8.303153144335519193479190862263
