| L(s) = 1 | + 2·2-s − 2·3-s + 4-s − 5-s − 4·6-s − 3·7-s − 2·8-s + 2·9-s − 2·10-s − 2·12-s + 6·13-s − 6·14-s + 2·15-s − 3·16-s + 4·17-s + 4·18-s − 3·19-s − 20-s + 6·21-s + 7·23-s + 4·24-s − 2·25-s + 12·26-s − 6·27-s − 3·28-s + 4·30-s − 12·31-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 1/2·4-s − 0.447·5-s − 1.63·6-s − 1.13·7-s − 0.707·8-s + 2/3·9-s − 0.632·10-s − 0.577·12-s + 1.66·13-s − 1.60·14-s + 0.516·15-s − 3/4·16-s + 0.970·17-s + 0.942·18-s − 0.688·19-s − 0.223·20-s + 1.30·21-s + 1.45·23-s + 0.816·24-s − 2/5·25-s + 2.35·26-s − 1.15·27-s − 0.566·28-s + 0.730·30-s − 2.15·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100010 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100010 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_2$ | \( ( 1 - T )( 1 - T + p T^{2} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 10 T + p T^{2} ) \) |
| 137 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 10 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 6 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 - 7 T + 54 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + T - 6 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 46 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 12 T + 78 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 7 T - 2 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 9 T + 98 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 10 T + 186 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 14 T + 138 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
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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
−14.1973781939, −13.5587098088, −13.3999252794, −12.9204191794, −12.5711164827, −12.2858800557, −11.8347267009, −11.1926545050, −10.8483871686, −10.7136811850, −9.80928270711, −9.33764088721, −8.99047504569, −8.32119659927, −7.59634248299, −7.13137869003, −6.40421513990, −6.04341218414, −5.75739641535, −5.22492561675, −4.54863271232, −3.97062137697, −3.37180621455, −3.14865063672, −1.50077266168, 0,
1.50077266168, 3.14865063672, 3.37180621455, 3.97062137697, 4.54863271232, 5.22492561675, 5.75739641535, 6.04341218414, 6.40421513990, 7.13137869003, 7.59634248299, 8.32119659927, 8.99047504569, 9.33764088721, 9.80928270711, 10.7136811850, 10.8483871686, 11.1926545050, 11.8347267009, 12.2858800557, 12.5711164827, 12.9204191794, 13.3999252794, 13.5587098088, 14.1973781939
