| L(s) = 1 | − 24·11-s + 104·31-s + 120·41-s + 8·61-s − 480·71-s − 9·81-s − 48·101-s − 124·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | − 2.18·11-s + 3.35·31-s + 2.92·41-s + 8/61·61-s − 6.76·71-s − 1/9·81-s − 0.475·101-s − 1.02·121-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 + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + 0.00440·227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.06398477399\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06398477399\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^3$ | \( 1 + 2594 T^{4} + p^{8} T^{8} \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p^{2} T^{2} )^{4} \) |
| 13 | $C_2^3$ | \( 1 - 55678 T^{4} + p^{8} T^{8} \) |
| 17 | $C_2^3$ | \( 1 - 145726 T^{4} + p^{8} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 622 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 110782 T^{4} + p^{8} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 622 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )^{4} \) |
| 37 | $C_2^3$ | \( 1 - 3244222 T^{4} + p^{8} T^{8} \) |
| 41 | $C_2$ | \( ( 1 - 30 T + p^{2} T^{2} )^{4} \) |
| 43 | $C_2^3$ | \( 1 - 2956702 T^{4} + p^{8} T^{8} \) |
| 47 | $C_2^3$ | \( 1 + 6128834 T^{4} + p^{8} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 11113634 T^{4} + p^{8} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 878 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^3$ | \( 1 - 39552286 T^{4} + p^{8} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + 120 T + p^{2} T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 - 46466686 T^{4} + p^{8} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 7006 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 2895458 T^{4} + p^{8} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 6658 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^3$ | \( 1 + 175254338 T^{4} + p^{8} T^{8} \) |
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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
−6.55407344131615032576316586103, −6.49994079638653908731622960205, −6.39542026536288844496662099382, −6.07171337603881759524851831416, −5.96511496822532849812780138200, −5.61145443792083203660057313036, −5.32050471397772106590872607226, −5.27388554708126221264501896748, −5.08248395027398204444576752464, −4.63048264281876853914591475028, −4.37895707179509980861773323512, −4.35477384833034909998548394738, −4.11668315582681139257306786202, −3.91443760724332416620707391183, −3.25907834594960316842357711469, −3.01160205768175695465272359305, −2.95335267150158216090052527296, −2.60014639425672334863075111385, −2.53269573771362903882498962722, −2.30653915746257481772251066723, −1.54817453605242963864031585183, −1.50096540507825417849013386941, −0.988993051327455225914769412872, −0.68144418631172377052370597822, −0.03997474052942285267603031522,
0.03997474052942285267603031522, 0.68144418631172377052370597822, 0.988993051327455225914769412872, 1.50096540507825417849013386941, 1.54817453605242963864031585183, 2.30653915746257481772251066723, 2.53269573771362903882498962722, 2.60014639425672334863075111385, 2.95335267150158216090052527296, 3.01160205768175695465272359305, 3.25907834594960316842357711469, 3.91443760724332416620707391183, 4.11668315582681139257306786202, 4.35477384833034909998548394738, 4.37895707179509980861773323512, 4.63048264281876853914591475028, 5.08248395027398204444576752464, 5.27388554708126221264501896748, 5.32050471397772106590872607226, 5.61145443792083203660057313036, 5.96511496822532849812780138200, 6.07171337603881759524851831416, 6.39542026536288844496662099382, 6.49994079638653908731622960205, 6.55407344131615032576316586103
