| L(s) = 1 | + 4·7-s − 9-s − 4·11-s − 2·13-s − 8·19-s + 4·23-s − 6·25-s − 10·29-s − 10·41-s − 8·43-s + 2·49-s − 4·63-s − 4·67-s + 6·73-s − 16·77-s + 8·79-s − 8·81-s + 4·83-s − 8·91-s + 4·99-s + 12·101-s + 28·103-s + 2·117-s − 6·121-s + 127-s + 131-s − 32·133-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 1/3·9-s − 1.20·11-s − 0.554·13-s − 1.83·19-s + 0.834·23-s − 6/5·25-s − 1.85·29-s − 1.56·41-s − 1.21·43-s + 2/7·49-s − 0.503·63-s − 0.488·67-s + 0.702·73-s − 1.82·77-s + 0.900·79-s − 8/9·81-s + 0.439·83-s − 0.838·91-s + 0.402·99-s + 1.19·101-s + 2.75·103-s + 0.184·117-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s − 2.77·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 23 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 39 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 65 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 65 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 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
−8.835049907303509484549593798231, −8.719551111858746293003092239488, −7.925637501151835379472346718459, −7.903441716070505716679739774389, −7.30875103208051269942178604231, −6.64940781179093504163969594519, −6.05859317323856151904262337277, −5.36042575580853315089921391866, −5.05371583429191354487143965316, −4.58771688728295132382625970009, −3.84996279867431347339299838894, −3.10646128265892577483267762074, −2.06868549011111082722180701691, −1.88480570955587165940305650397, 0,
1.88480570955587165940305650397, 2.06868549011111082722180701691, 3.10646128265892577483267762074, 3.84996279867431347339299838894, 4.58771688728295132382625970009, 5.05371583429191354487143965316, 5.36042575580853315089921391866, 6.05859317323856151904262337277, 6.64940781179093504163969594519, 7.30875103208051269942178604231, 7.903441716070505716679739774389, 7.925637501151835379472346718459, 8.719551111858746293003092239488, 8.835049907303509484549593798231
