| L(s) = 1 | − 2-s − 4-s + 5-s + 7-s + 3·8-s − 10-s − 11-s − 6·13-s − 14-s − 16-s + 3·19-s − 20-s + 22-s − 6·23-s − 4·25-s + 6·26-s − 28-s + 3·29-s + 3·31-s − 5·32-s + 35-s − 3·37-s − 3·38-s + 3·40-s − 9·41-s + 12·43-s + 44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.447·5-s + 0.377·7-s + 1.06·8-s − 0.316·10-s − 0.301·11-s − 1.66·13-s − 0.267·14-s − 1/4·16-s + 0.688·19-s − 0.223·20-s + 0.213·22-s − 1.25·23-s − 4/5·25-s + 1.17·26-s − 0.188·28-s + 0.557·29-s + 0.538·31-s − 0.883·32-s + 0.169·35-s − 0.493·37-s − 0.486·38-s + 0.474·40-s − 1.40·41-s + 1.82·43-s + 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $D_{4}$ | \( 1 - 3 T + 11 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 24 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 3 T + 45 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 3 T - 10 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 3 T + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 + 9 T + 71 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 12 T + 114 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 9 T + 60 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 15 T + 144 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 61 | $D_{4}$ | \( 1 - 9 T + 85 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6 T + 78 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 9 T + 38 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 13 T + 152 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 9 T + 13 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 7 T + 50 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 15 T + 196 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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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
−14.0122472296, −13.5813711815, −13.3577283094, −12.7245023112, −12.1448645222, −11.9329751953, −11.5249985835, −10.7182818030, −10.3177386940, −9.95805760848, −9.79415120324, −9.13333075399, −8.74235259852, −8.17433822836, −7.73329220727, −7.36836907953, −6.89674247310, −6.04555917057, −5.50523449902, −5.08250740306, −4.43932935713, −4.01418345425, −2.95128444225, −2.25203657437, −1.43437250182, 0,
1.43437250182, 2.25203657437, 2.95128444225, 4.01418345425, 4.43932935713, 5.08250740306, 5.50523449902, 6.04555917057, 6.89674247310, 7.36836907953, 7.73329220727, 8.17433822836, 8.74235259852, 9.13333075399, 9.79415120324, 9.95805760848, 10.3177386940, 10.7182818030, 11.5249985835, 11.9329751953, 12.1448645222, 12.7245023112, 13.3577283094, 13.5813711815, 14.0122472296
