L(s) = 1 | + 2-s + 4-s + 2·7-s + 8-s + 9-s + 4·11-s + 2·14-s + 16-s + 18-s + 4·22-s − 8·23-s − 25-s + 2·28-s + 8·29-s + 32-s + 36-s + 8·37-s + 8·43-s + 4·44-s − 8·46-s − 3·49-s − 50-s − 4·53-s + 2·56-s + 8·58-s + 2·63-s + 64-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.755·7-s + 0.353·8-s + 1/3·9-s + 1.20·11-s + 0.534·14-s + 1/4·16-s + 0.235·18-s + 0.852·22-s − 1.66·23-s − 1/5·25-s + 0.377·28-s + 1.48·29-s + 0.176·32-s + 1/6·36-s + 1.31·37-s + 1.21·43-s + 0.603·44-s − 1.17·46-s − 3/7·49-s − 0.141·50-s − 0.549·53-s + 0.267·56-s + 1.05·58-s + 0.251·63-s + 1/8·64-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
\(L(1)\) |
\(\approx\) |
\(4.103484619\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.103484619\) |
\(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
good | 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 114 T^{2} + p^{2} T^{4} \) |
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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
−8.217517574338029881047298024315, −7.80450097971650473473473987567, −7.49418306301816971548523252525, −6.83884082190614807963866344137, −6.42488746844684560222654732178, −6.06130615371734066713217062980, −5.68357557756903974786343710768, −4.88140459649061728484536054139, −4.59418104424041763449519389564, −4.08585233157219342575261319389, −3.75629949933532880979224361663, −2.97488482743920153177966076036, −2.30470728397877880314677912481, −1.69381180944090386118854239481, −0.965723605583331135043984603230,
0.965723605583331135043984603230, 1.69381180944090386118854239481, 2.30470728397877880314677912481, 2.97488482743920153177966076036, 3.75629949933532880979224361663, 4.08585233157219342575261319389, 4.59418104424041763449519389564, 4.88140459649061728484536054139, 5.68357557756903974786343710768, 6.06130615371734066713217062980, 6.42488746844684560222654732178, 6.83884082190614807963866344137, 7.49418306301816971548523252525, 7.80450097971650473473473987567, 8.217517574338029881047298024315