| L(s) = 1 | − 3·9-s + 2·11-s − 6·13-s − 17-s + 5·19-s − 23-s + 25-s − 2·29-s − 3·31-s + 2·43-s + 4·47-s − 6·49-s − 5·53-s + 8·59-s − 13·61-s + 2·67-s − 18·71-s + 4·73-s − 5·79-s + 9·81-s − 7·83-s + 4·89-s + 2·97-s − 6·99-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 9-s + 0.603·11-s − 1.66·13-s − 0.242·17-s + 1.14·19-s − 0.208·23-s + 1/5·25-s − 0.371·29-s − 0.538·31-s + 0.304·43-s + 0.583·47-s − 6/7·49-s − 0.686·53-s + 1.04·59-s − 1.66·61-s + 0.244·67-s − 2.13·71-s + 0.468·73-s − 0.562·79-s + 81-s − 0.768·83-s + 0.423·89-s + 0.203·97-s − 0.603·99-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 172800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 172800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 + T + 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 5 T + 18 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + T + 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $D_{4}$ | \( 1 + 3 T + 6 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 5 T + 96 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 86 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 13 T + 128 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $D_{4}$ | \( 1 + 18 T + 206 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 5 T + 126 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 7 T + 38 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $D_{4}$ | \( 1 - 2 T + 130 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−13.7478259492, −13.4556404946, −12.8469125071, −12.4141549360, −11.9523734410, −11.7689126728, −11.2690579257, −10.8002206856, −10.3241510927, −9.73805306794, −9.36553216910, −9.11896767185, −8.47770528385, −7.97454150864, −7.42086627953, −7.16848110760, −6.49838386536, −5.93388353220, −5.42348995862, −4.96968708740, −4.37658117153, −3.65127172166, −2.94550230062, −2.47346227497, −1.47179819197, 0,
1.47179819197, 2.47346227497, 2.94550230062, 3.65127172166, 4.37658117153, 4.96968708740, 5.42348995862, 5.93388353220, 6.49838386536, 7.16848110760, 7.42086627953, 7.97454150864, 8.47770528385, 9.11896767185, 9.36553216910, 9.73805306794, 10.3241510927, 10.8002206856, 11.2690579257, 11.7689126728, 11.9523734410, 12.4141549360, 12.8469125071, 13.4556404946, 13.7478259492
