| L(s) = 1 | − 6·7-s + 14·17-s + 8·23-s + 25-s + 16·31-s − 4·41-s − 14·47-s + 13·49-s − 6·71-s + 28·73-s + 20·79-s + 16·97-s + 12·103-s + 12·113-s − 84·119-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 48·161-s + 163-s + 167-s − 169-s + ⋯ |
| L(s) = 1 | − 2.26·7-s + 3.39·17-s + 1.66·23-s + 1/5·25-s + 2.87·31-s − 0.624·41-s − 2.04·47-s + 13/7·49-s − 0.712·71-s + 3.27·73-s + 2.25·79-s + 1.62·97-s + 1.18·103-s + 1.12·113-s − 7.70·119-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s − 3.78·161-s + 0.0783·163-s + 0.0773·167-s − 0.0769·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14017536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14017536 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.762532128\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.762532128\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−8.634062321997018886648688134846, −8.249013359938070707737154342917, −8.024390805779863929082350778109, −7.58448174844116243181029221418, −7.18313079836657899187790598726, −6.78827086121344587444916827575, −6.36091404248984226508671355346, −6.24822188770034146979853787686, −5.89642221190825359384461806898, −5.24572452337337432413576212737, −4.85360871593317142099655212774, −4.83755644608983131410532151537, −3.66409819739302987864936745574, −3.63313328700375840805774317030, −3.11027091550498953658062136111, −3.06833981906607753311022033292, −2.50667611440250328149732812498, −1.61379954473299415405957426706, −0.814596216461762226005109417355, −0.72156226730330465486695067466,
0.72156226730330465486695067466, 0.814596216461762226005109417355, 1.61379954473299415405957426706, 2.50667611440250328149732812498, 3.06833981906607753311022033292, 3.11027091550498953658062136111, 3.63313328700375840805774317030, 3.66409819739302987864936745574, 4.83755644608983131410532151537, 4.85360871593317142099655212774, 5.24572452337337432413576212737, 5.89642221190825359384461806898, 6.24822188770034146979853787686, 6.36091404248984226508671355346, 6.78827086121344587444916827575, 7.18313079836657899187790598726, 7.58448174844116243181029221418, 8.024390805779863929082350778109, 8.249013359938070707737154342917, 8.634062321997018886648688134846
