| L(s) = 1 | + 2·2-s + 2·4-s + 4·8-s + 6·9-s + 18·13-s + 8·16-s − 18·17-s + 12·18-s − 4·25-s + 36·26-s − 10·29-s + 8·32-s − 36·34-s + 12·36-s − 6·37-s + 6·41-s − 2·49-s − 8·50-s + 36·52-s − 2·53-s − 20·58-s + 18·61-s + 8·64-s − 36·68-s + 24·72-s − 42·73-s − 12·74-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 1.41·8-s + 2·9-s + 4.99·13-s + 2·16-s − 4.36·17-s + 2.82·18-s − 4/5·25-s + 7.06·26-s − 1.85·29-s + 1.41·32-s − 6.17·34-s + 2·36-s − 0.986·37-s + 0.937·41-s − 2/7·49-s − 1.13·50-s + 4.99·52-s − 0.274·53-s − 2.62·58-s + 2.30·61-s + 64-s − 4.36·68-s + 2.82·72-s − 4.91·73-s − 1.39·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.747182695\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.747182695\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 9 T + 44 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^3$ | \( 1 - 35 T^{2} + 864 T^{4} - 35 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 5 T - 4 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^3$ | \( 1 - 35 T^{2} + 264 T^{4} - 35 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 3 T + 44 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 - 35 T^{2} - 624 T^{4} - 35 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^3$ | \( 1 - 91 T^{2} + 6072 T^{4} - 91 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + T - 52 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 115 T^{2} + 9744 T^{4} - 115 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 9 T + 88 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 53 T^{2} - 1680 T^{4} + 53 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 138 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 21 T + 220 T^{2} + 21 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^3$ | \( 1 + 149 T^{2} + 15960 T^{4} + 149 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $C_2^3$ | \( 1 - 139 T^{2} + 12432 T^{4} - 139 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 15 T + 164 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 3 T + 100 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.708761099332485275511802288944, −8.587047919982227279923810459854, −8.268518419870145807628433761855, −8.193559127553857657135930903240, −7.46964435319284695911721295976, −7.32300948998631258951847966136, −7.23944031512135993350236346880, −6.82887323972644808475642856110, −6.56952448695260347795331798014, −6.18706567605883046553596881412, −6.06556071043135975665215796846, −6.06009229468338389542901405614, −5.58876086320798678931715335107, −5.04324386480462238592226540436, −4.66136517922010479613159921149, −4.64204744619510715749249436556, −4.04807324126781358927505557863, −3.92342805970372108328068343570, −3.81435655659188614688426993024, −3.72957285996390457444583341717, −3.09182074643682367408598736002, −2.27254283767654302445476477820, −1.93478447149307208096553483780, −1.57616325475670918537306844807, −1.22056919999135260251214175857,
1.22056919999135260251214175857, 1.57616325475670918537306844807, 1.93478447149307208096553483780, 2.27254283767654302445476477820, 3.09182074643682367408598736002, 3.72957285996390457444583341717, 3.81435655659188614688426993024, 3.92342805970372108328068343570, 4.04807324126781358927505557863, 4.64204744619510715749249436556, 4.66136517922010479613159921149, 5.04324386480462238592226540436, 5.58876086320798678931715335107, 6.06009229468338389542901405614, 6.06556071043135975665215796846, 6.18706567605883046553596881412, 6.56952448695260347795331798014, 6.82887323972644808475642856110, 7.23944031512135993350236346880, 7.32300948998631258951847966136, 7.46964435319284695911721295976, 8.193559127553857657135930903240, 8.268518419870145807628433761855, 8.587047919982227279923810459854, 8.708761099332485275511802288944
