| L(s) = 1 | − 7-s − 2·11-s + 10·13-s − 2·17-s + 3·19-s − 2·23-s + 5·25-s − 16·29-s − 31-s + 5·37-s − 4·41-s − 14·43-s + 8·47-s − 6·49-s − 2·53-s − 10·59-s + 2·61-s − 11·67-s + 24·71-s + 3·73-s + 2·77-s + 17·79-s − 32·83-s + 12·89-s − 10·91-s − 28·97-s + 18·101-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 0.603·11-s + 2.77·13-s − 0.485·17-s + 0.688·19-s − 0.417·23-s + 25-s − 2.97·29-s − 0.179·31-s + 0.821·37-s − 0.624·41-s − 2.13·43-s + 1.16·47-s − 6/7·49-s − 0.274·53-s − 1.30·59-s + 0.256·61-s − 1.34·67-s + 2.84·71-s + 0.351·73-s + 0.227·77-s + 1.91·79-s − 3.51·83-s + 1.27·89-s − 1.04·91-s − 2.84·97-s + 1.79·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.855855644\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.855855644\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 3 T - 10 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 2 T - 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + T - 30 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 5 T - 12 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 2 T - 49 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 10 T + 41 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 3 T - 64 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 12 T + 55 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−9.321373833852006787912146243776, −8.999333564923491531527365074893, −8.534248896308258362089089932393, −8.087817976670662742418047574168, −8.048392526185724101371082483233, −7.35562504577457230625684515014, −6.93641609323326693702872909512, −6.62864834393501251245400976408, −6.10808034512077537490277602006, −5.74853529501354851062450096635, −5.56248223166213060823120987528, −4.89757100787011739868456377650, −4.48280947861078819784756692361, −3.75131724882290840631377320753, −3.54599306599638251290323240384, −3.27656771540192437820912824979, −2.52824361942447923571916898319, −1.74243995647585660642647000863, −1.45543627586309498611810570685, −0.49884445031552276713561240693,
0.49884445031552276713561240693, 1.45543627586309498611810570685, 1.74243995647585660642647000863, 2.52824361942447923571916898319, 3.27656771540192437820912824979, 3.54599306599638251290323240384, 3.75131724882290840631377320753, 4.48280947861078819784756692361, 4.89757100787011739868456377650, 5.56248223166213060823120987528, 5.74853529501354851062450096635, 6.10808034512077537490277602006, 6.62864834393501251245400976408, 6.93641609323326693702872909512, 7.35562504577457230625684515014, 8.048392526185724101371082483233, 8.087817976670662742418047574168, 8.534248896308258362089089932393, 8.999333564923491531527365074893, 9.321373833852006787912146243776
