| L(s) = 1 | + 5-s − 4·7-s + 9-s − 2·13-s − 2·17-s + 4·19-s − 4·23-s + 25-s + 8·29-s − 4·31-s − 4·35-s − 2·37-s + 8·43-s + 45-s − 4·47-s + 2·49-s + 2·53-s − 16·61-s − 4·63-s − 2·65-s + 8·67-s − 16·71-s − 14·73-s + 4·79-s + 81-s − 2·85-s + 8·91-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.51·7-s + 1/3·9-s − 0.554·13-s − 0.485·17-s + 0.917·19-s − 0.834·23-s + 1/5·25-s + 1.48·29-s − 0.718·31-s − 0.676·35-s − 0.328·37-s + 1.21·43-s + 0.149·45-s − 0.583·47-s + 2/7·49-s + 0.274·53-s − 2.04·61-s − 0.503·63-s − 0.248·65-s + 0.977·67-s − 1.89·71-s − 1.63·73-s + 0.450·79-s + 1/9·81-s − 0.216·85-s + 0.838·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$ | \( 1 - T \) |
| good | 7 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 58 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $D_{4}$ | \( 1 + 16 T + 158 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−13.2485869999, −12.9047441355, −12.4907543972, −12.2292076934, −11.7153320768, −11.3085796016, −10.5736288497, −10.3198018442, −9.96268087321, −9.55128340376, −9.11287509262, −8.87249891367, −8.07624390394, −7.66105034770, −7.09587128794, −6.75812298191, −6.20954917528, −5.88784386914, −5.28467115822, −4.61019396376, −4.17218661587, −3.34104755663, −2.96648130022, −2.29330400965, −1.33490176948, 0,
1.33490176948, 2.29330400965, 2.96648130022, 3.34104755663, 4.17218661587, 4.61019396376, 5.28467115822, 5.88784386914, 6.20954917528, 6.75812298191, 7.09587128794, 7.66105034770, 8.07624390394, 8.87249891367, 9.11287509262, 9.55128340376, 9.96268087321, 10.3198018442, 10.5736288497, 11.3085796016, 11.7153320768, 12.2292076934, 12.4907543972, 12.9047441355, 13.2485869999
