L(s) = 1 | − 2-s + 3-s + 4-s + 3·5-s − 6-s − 8-s − 2·9-s − 3·10-s + 12-s + 3·15-s + 16-s + 2·18-s − 2·19-s + 3·20-s − 6·23-s − 24-s − 25-s − 5·27-s − 12·29-s − 3·30-s − 32-s − 2·36-s + 2·38-s − 3·40-s + 7·43-s − 6·45-s + 6·46-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.34·5-s − 0.408·6-s − 0.353·8-s − 2/3·9-s − 0.948·10-s + 0.288·12-s + 0.774·15-s + 1/4·16-s + 0.471·18-s − 0.458·19-s + 0.670·20-s − 1.25·23-s − 0.204·24-s − 1/5·25-s − 0.962·27-s − 2.22·29-s − 0.547·30-s − 0.176·32-s − 1/3·36-s + 0.324·38-s − 0.474·40-s + 1.06·43-s − 0.894·45-s + 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 5 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 80 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 113 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 61 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−8.651707926225404781781052275806, −7.981825706181436972776941879022, −7.86045522688470142906013735097, −7.28949864139311801546943708983, −6.61090493813459854147139575588, −6.08251093950055890212735051070, −5.82024171278327378828822460508, −5.44491032950393586920797555705, −4.64782609107135243902649102294, −3.83578581819055462071722812434, −3.40143729930981129634384981430, −2.52882026207307242437407028298, −2.06227410289181150586814080815, −1.62901038406687568185313451564, 0,
1.62901038406687568185313451564, 2.06227410289181150586814080815, 2.52882026207307242437407028298, 3.40143729930981129634384981430, 3.83578581819055462071722812434, 4.64782609107135243902649102294, 5.44491032950393586920797555705, 5.82024171278327378828822460508, 6.08251093950055890212735051070, 6.61090493813459854147139575588, 7.28949864139311801546943708983, 7.86045522688470142906013735097, 7.981825706181436972776941879022, 8.651707926225404781781052275806