| L(s) = 1 | − 2·2-s − 2·3-s + 3·4-s + 4·6-s − 4·8-s + 3·9-s + 2·11-s − 6·12-s − 4·13-s + 5·16-s − 6·18-s − 4·19-s − 4·22-s + 8·24-s + 2·25-s + 8·26-s − 4·27-s − 12·29-s − 4·31-s − 6·32-s − 4·33-s + 9·36-s + 4·37-s + 8·38-s + 8·39-s − 8·43-s + 6·44-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s + 1.63·6-s − 1.41·8-s + 9-s + 0.603·11-s − 1.73·12-s − 1.10·13-s + 5/4·16-s − 1.41·18-s − 0.917·19-s − 0.852·22-s + 1.63·24-s + 2/5·25-s + 1.56·26-s − 0.769·27-s − 2.22·29-s − 0.718·31-s − 1.06·32-s − 0.696·33-s + 3/2·36-s + 0.657·37-s + 1.29·38-s + 1.28·39-s − 1.21·43-s + 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10458756 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10458756 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 12 T + 118 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 12 T + 94 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 150 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 94 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−8.636654573560322787009352770518, −7.980839856472816231512541391835, −7.73075203722081899979590398312, −7.34194120884871951220201843611, −7.04724458323073659643565916361, −6.71472124242437005174465484699, −6.20055703995580046933943628098, −6.12233094968129953908267084798, −5.33772997840573494967215079970, −5.30632100621679819804951013256, −4.69911310563795507840422267558, −4.20554096578801022587513917028, −3.68522956008377636478202976477, −3.30513755452804215816602600365, −2.47323405734273934052503713735, −2.10433451364582797321674811320, −1.59901655424549639312613307082, −1.04983271370502453256593629120, 0, 0,
1.04983271370502453256593629120, 1.59901655424549639312613307082, 2.10433451364582797321674811320, 2.47323405734273934052503713735, 3.30513755452804215816602600365, 3.68522956008377636478202976477, 4.20554096578801022587513917028, 4.69911310563795507840422267558, 5.30632100621679819804951013256, 5.33772997840573494967215079970, 6.12233094968129953908267084798, 6.20055703995580046933943628098, 6.71472124242437005174465484699, 7.04724458323073659643565916361, 7.34194120884871951220201843611, 7.73075203722081899979590398312, 7.980839856472816231512541391835, 8.636654573560322787009352770518
