L(s) = 1 | − 2·3-s − 2·7-s + 9-s − 8·17-s + 4·21-s + 25-s + 4·27-s + 12·37-s + 4·41-s − 4·43-s − 4·47-s − 3·49-s + 16·51-s − 4·59-s − 2·63-s − 4·67-s − 2·75-s − 11·81-s − 4·83-s − 20·89-s + 4·109-s − 24·111-s + 16·119-s + 18·121-s − 8·123-s + 127-s + 8·129-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 0.755·7-s + 1/3·9-s − 1.94·17-s + 0.872·21-s + 1/5·25-s + 0.769·27-s + 1.97·37-s + 0.624·41-s − 0.609·43-s − 0.583·47-s − 3/7·49-s + 2.24·51-s − 0.520·59-s − 0.251·63-s − 0.488·67-s − 0.230·75-s − 1.22·81-s − 0.439·83-s − 2.11·89-s + 0.383·109-s − 2.27·111-s + 1.46·119-s + 1.63·121-s − 0.721·123-s + 0.0887·127-s + 0.704·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5218431217\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5218431217\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
good | 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 114 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 106 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.408117425151904216496611450372, −7.82699557987974965500454193081, −7.24844436608886519314565632809, −6.82702054251538257784036372229, −6.45028970478071764771716167149, −6.07947035343431949116623572903, −5.77094730638512843812633272466, −5.07347122617894016164139827506, −4.56175599520419310312532155307, −4.33605440512239448579716628130, −3.56533580319244605053018339486, −2.84312058509055079259451948968, −2.41135331513060301566509708235, −1.43459398945060066541016937959, −0.38880483690197515807465874652,
0.38880483690197515807465874652, 1.43459398945060066541016937959, 2.41135331513060301566509708235, 2.84312058509055079259451948968, 3.56533580319244605053018339486, 4.33605440512239448579716628130, 4.56175599520419310312532155307, 5.07347122617894016164139827506, 5.77094730638512843812633272466, 6.07947035343431949116623572903, 6.45028970478071764771716167149, 6.82702054251538257784036372229, 7.24844436608886519314565632809, 7.82699557987974965500454193081, 8.408117425151904216496611450372