| L(s) = 1 | − 3·2-s − 3·3-s + 4·4-s − 6·5-s + 9·6-s − 6·7-s − 3·8-s + 4·9-s + 18·10-s − 4·11-s − 12·12-s − 5·13-s + 18·14-s + 18·15-s + 3·16-s − 4·17-s − 12·18-s − 5·19-s − 24·20-s + 18·21-s + 12·22-s − 8·23-s + 9·24-s + 18·25-s + 15·26-s − 24·28-s − 29-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.73·3-s + 2·4-s − 2.68·5-s + 3.67·6-s − 2.26·7-s − 1.06·8-s + 4/3·9-s + 5.69·10-s − 1.20·11-s − 3.46·12-s − 1.38·13-s + 4.81·14-s + 4.64·15-s + 3/4·16-s − 0.970·17-s − 2.82·18-s − 1.14·19-s − 5.36·20-s + 3.92·21-s + 2.55·22-s − 1.66·23-s + 1.83·24-s + 18/5·25-s + 2.94·26-s − 4.53·28-s − 0.185·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39993 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39993 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| 13331 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 70 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $D_{4}$ | \( 1 + 6 T + 3 p T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 5 T + 24 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 + 8 T + 50 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + T - 15 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 3 T - 5 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 3 T - 27 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 72 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - T - 60 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 34 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 7 T + 60 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 109 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - T - 47 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 17 T + 163 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T - 26 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 9 T + 113 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 18 T + 197 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 4 p T^{3} + p^{2} T^{4} \) |
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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
−15.9704645837, −15.6693054009, −15.2056094600, −14.7641403995, −13.6456655209, −12.8592480806, −12.5953132339, −12.3023007929, −11.7010824178, −11.5251462003, −10.8563797264, −10.3700922436, −10.0207472259, −9.79274056879, −8.89574331067, −8.38228702161, −8.08029649442, −7.37677982037, −7.08567270792, −6.48947692440, −5.97422717765, −4.98342638789, −4.31718290768, −3.59646164255, −2.70281332604, 0, 0, 0,
2.70281332604, 3.59646164255, 4.31718290768, 4.98342638789, 5.97422717765, 6.48947692440, 7.08567270792, 7.37677982037, 8.08029649442, 8.38228702161, 8.89574331067, 9.79274056879, 10.0207472259, 10.3700922436, 10.8563797264, 11.5251462003, 11.7010824178, 12.3023007929, 12.5953132339, 12.8592480806, 13.6456655209, 14.7641403995, 15.2056094600, 15.6693054009, 15.9704645837
