| L(s) = 1 | − 20·5-s − 8·9-s + 16·11-s − 20·19-s + 250·25-s + 160·45-s + 4·49-s − 320·55-s + 400·61-s − 114·81-s + 400·95-s − 128·99-s + 16·101-s − 324·121-s − 2.50e3·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 424·169-s + 160·171-s + 173-s + ⋯ |
| L(s) = 1 | − 4·5-s − 8/9·9-s + 1.45·11-s − 1.05·19-s + 10·25-s + 32/9·45-s + 4/49·49-s − 5.81·55-s + 6.55·61-s − 1.40·81-s + 4.21·95-s − 1.29·99-s + 0.158·101-s − 2.67·121-s − 20·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 2.50·169-s + 0.935·171-s + 0.00578·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6222740427\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6222740427\) |
| \(L(2)\) |
|
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 | $C_1$ | \( ( 1 + p T )^{4} \) |
| 19 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| good | 3 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2}( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p^{2} T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 + 212 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 194 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 962 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 338 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 50 T + p^{2} T^{2} )^{2}( 1 + 50 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 1604 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 2018 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 1006 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 4322 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 2468 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 1586 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 100 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 8852 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 8738 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 10274 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 386 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 12914 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 5662 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 3572 T^{2} + p^{4} T^{4} )^{2} \) |
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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
−8.028384187159167404224475557036, −7.953810778608334633139076087399, −7.57158604575284248828644447285, −7.16462060014979498520772606038, −6.96958933114405132402230360969, −6.91111301432792060720032354238, −6.69300177185083550730662522146, −6.47230037347104298823465827339, −6.04450655164885400936240255809, −5.51332006375259276678092410948, −5.34082048860877066618963102938, −5.17316280045941316953517704142, −4.51461617531968255687875129590, −4.50935786954804357446592381000, −4.12808239585791544690296633834, −3.94371164061317452962283155415, −3.82225869341339774614690558526, −3.42269381109924929813184693628, −3.21085231723320378163872647155, −2.79074090318952672253101797640, −2.45048175229460902300760867628, −1.86660518366274721539016618816, −0.961598945704208431278308427858, −0.811278483283539593880593572809, −0.26818955682233532850469202854,
0.26818955682233532850469202854, 0.811278483283539593880593572809, 0.961598945704208431278308427858, 1.86660518366274721539016618816, 2.45048175229460902300760867628, 2.79074090318952672253101797640, 3.21085231723320378163872647155, 3.42269381109924929813184693628, 3.82225869341339774614690558526, 3.94371164061317452962283155415, 4.12808239585791544690296633834, 4.50935786954804357446592381000, 4.51461617531968255687875129590, 5.17316280045941316953517704142, 5.34082048860877066618963102938, 5.51332006375259276678092410948, 6.04450655164885400936240255809, 6.47230037347104298823465827339, 6.69300177185083550730662522146, 6.91111301432792060720032354238, 6.96958933114405132402230360969, 7.16462060014979498520772606038, 7.57158604575284248828644447285, 7.953810778608334633139076087399, 8.028384187159167404224475557036
