| L(s) = 1 | − 4·3-s + 6·9-s − 12·17-s − 8·19-s + 25-s + 4·27-s + 12·41-s − 20·43-s − 10·49-s + 48·51-s + 32·57-s + 24·59-s + 4·67-s + 4·73-s − 4·75-s − 37·81-s + 12·83-s − 12·89-s + 4·97-s − 12·107-s − 12·113-s − 22·121-s − 48·123-s + 127-s + 80·129-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 2·9-s − 2.91·17-s − 1.83·19-s + 1/5·25-s + 0.769·27-s + 1.87·41-s − 3.04·43-s − 1.42·49-s + 6.72·51-s + 4.23·57-s + 3.12·59-s + 0.488·67-s + 0.468·73-s − 0.461·75-s − 4.11·81-s + 1.31·83-s − 1.27·89-s + 0.406·97-s − 1.16·107-s − 1.12·113-s − 2·121-s − 4.32·123-s + 0.0887·127-s + 7.04·129-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−11.50854453196989171707939391081, −11.16454648418111686213251777526, −10.83473950301065015207539180976, −10.35239107507603235483123434521, −9.511328653906225471558818278712, −8.552217550781204179646223792184, −8.404829898294941223649273971532, −6.87772159023272676690262517558, −6.57891116465648258947670054106, −6.26903409609752801893873868553, −5.32703134203753576211066117067, −4.78130792717525308450176413839, −4.13045861856120535222472022668, −2.32736297473362010207341859714, 0,
2.32736297473362010207341859714, 4.13045861856120535222472022668, 4.78130792717525308450176413839, 5.32703134203753576211066117067, 6.26903409609752801893873868553, 6.57891116465648258947670054106, 6.87772159023272676690262517558, 8.404829898294941223649273971532, 8.552217550781204179646223792184, 9.511328653906225471558818278712, 10.35239107507603235483123434521, 10.83473950301065015207539180976, 11.16454648418111686213251777526, 11.50854453196989171707939391081
