| L(s) = 1 | − 3-s − 8·7-s − 2·9-s − 8·13-s + 2·19-s + 8·21-s + 6·25-s + 5·27-s − 4·31-s + 8·39-s − 4·43-s + 35·49-s − 2·57-s + 16·63-s − 6·67-s + 14·73-s − 6·75-s + 81-s + 64·91-s + 4·93-s + 20·97-s − 28·103-s + 16·117-s + 2·121-s + 127-s + 4·129-s + 131-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 3.02·7-s − 2/3·9-s − 2.21·13-s + 0.458·19-s + 1.74·21-s + 6/5·25-s + 0.962·27-s − 0.718·31-s + 1.28·39-s − 0.609·43-s + 5·49-s − 0.264·57-s + 2.01·63-s − 0.733·67-s + 1.63·73-s − 0.692·75-s + 1/9·81-s + 6.70·91-s + 0.414·93-s + 2.03·97-s − 2.75·103-s + 1.47·117-s + 2/11·121-s + 0.0887·127-s + 0.352·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 75 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 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
−7.926820183763721273408928219535, −7.27516876706279029677668253367, −7.04683860249327797983136220502, −6.61719661495160617371104725748, −6.36135315411272810583593262467, −5.74799582254727060198334650149, −5.38243383236233899214435113139, −4.91656109937170581336625223059, −4.30746972195725292340718685382, −3.49774119230959878062143732209, −3.12920508965279611598136191887, −2.79574488179296944944953834646, −2.18659980957160077689103512426, −0.63905617607425876272828327161, 0,
0.63905617607425876272828327161, 2.18659980957160077689103512426, 2.79574488179296944944953834646, 3.12920508965279611598136191887, 3.49774119230959878062143732209, 4.30746972195725292340718685382, 4.91656109937170581336625223059, 5.38243383236233899214435113139, 5.74799582254727060198334650149, 6.36135315411272810583593262467, 6.61719661495160617371104725748, 7.04683860249327797983136220502, 7.27516876706279029677668253367, 7.926820183763721273408928219535
