| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 7-s + 2·8-s − 10-s + 12-s + 14-s − 15-s + 2·16-s − 2·17-s − 20-s + 21-s + 2·24-s + 25-s − 27-s + 28-s − 30-s + 2·31-s + 2·32-s − 2·34-s − 35-s − 2·40-s − 41-s + 42-s + ⋯ |
| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 7-s + 2·8-s − 10-s + 12-s + 14-s − 15-s + 2·16-s − 2·17-s − 20-s + 21-s + 2·24-s + 25-s − 27-s + 28-s − 30-s + 2·31-s + 2·32-s − 2·34-s − 35-s − 2·40-s − 41-s + 42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1147041 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1147041 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.577484448\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.577484448\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 - T + T^{2} \) |
| 17 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
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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
−10.31000785960070973425145987885, −10.18314074088742088864122875315, −9.237785970730922843029212030175, −9.047279215728615495218799377281, −8.342942806059765623073148896806, −8.162956000536297151254109600692, −7.85716807480423479596624065581, −7.59740797799387411836843228304, −6.72752367581865303507938978190, −6.71777530235811452021515171274, −6.24989816431578657861024580565, −5.24555071645900999815214511114, −4.90316146182703296405034664977, −4.66354987104887455240327101398, −4.11161519989101039002058426099, −3.83049749101874292939163297482, −2.92325442414945108009616314273, −2.82531929453971743904694901829, −1.84929324275897511436347229686, −1.56784297692170006947758669836,
1.56784297692170006947758669836, 1.84929324275897511436347229686, 2.82531929453971743904694901829, 2.92325442414945108009616314273, 3.83049749101874292939163297482, 4.11161519989101039002058426099, 4.66354987104887455240327101398, 4.90316146182703296405034664977, 5.24555071645900999815214511114, 6.24989816431578657861024580565, 6.71777530235811452021515171274, 6.72752367581865303507938978190, 7.59740797799387411836843228304, 7.85716807480423479596624065581, 8.162956000536297151254109600692, 8.342942806059765623073148896806, 9.047279215728615495218799377281, 9.237785970730922843029212030175, 10.18314074088742088864122875315, 10.31000785960070973425145987885
