L(s) = 1 | − 12·5-s − 3·9-s + 28·13-s − 12·17-s + 58·25-s − 60·29-s − 52·37-s − 108·41-s + 36·45-s + 50·49-s + 36·53-s + 140·61-s − 336·65-s + 164·73-s + 9·81-s + 144·85-s + 228·89-s + 68·97-s + 36·101-s − 68·109-s − 156·113-s − 84·117-s − 190·121-s + 36·125-s + 127-s + 131-s + 137-s + ⋯ |
L(s) = 1 | − 2.39·5-s − 1/3·9-s + 2.15·13-s − 0.705·17-s + 2.31·25-s − 2.06·29-s − 1.40·37-s − 2.63·41-s + 4/5·45-s + 1.02·49-s + 0.679·53-s + 2.29·61-s − 5.16·65-s + 2.24·73-s + 1/9·81-s + 1.69·85-s + 2.56·89-s + 0.701·97-s + 0.356·101-s − 0.623·109-s − 1.38·113-s − 0.717·117-s − 1.57·121-s + 0.287·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7459479299\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7459479299\) |
\(L(2)\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
good | 5 | $C_2$ | \( ( 1 + 6 T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 50 T^{2} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 190 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 674 T^{2} + p^{4} T^{4} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 30 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 1490 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 26 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 54 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 3266 T^{2} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 2690 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 18 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 6530 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 70 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 4894 T^{2} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 3170 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 82 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 6674 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 13346 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 114 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 34 T + p^{2} T^{2} )^{2} \) |
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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
−12.31913464371876665447767818138, −11.88835318663780390861657362668, −11.61053766197120616313103490582, −11.18129072154113067986754908535, −10.81365970397423608373755426679, −10.36331169183046435625152979529, −9.400841679566366039215493598883, −8.816981222403210212250512039106, −8.377607145457580686486383334929, −8.188910495025699306118599680339, −7.49852036275646657176237460762, −6.95876608073337558873293685233, −6.48320625867505541434603991069, −5.60603373814645247556759848579, −5.02082383370425517343335080136, −4.00561473197111105332721914577, −3.66740012015501819603540992330, −3.47893809978424182834916222429, −1.93539103365095390518858657712, −0.50814040392929698989456080219,
0.50814040392929698989456080219, 1.93539103365095390518858657712, 3.47893809978424182834916222429, 3.66740012015501819603540992330, 4.00561473197111105332721914577, 5.02082383370425517343335080136, 5.60603373814645247556759848579, 6.48320625867505541434603991069, 6.95876608073337558873293685233, 7.49852036275646657176237460762, 8.188910495025699306118599680339, 8.377607145457580686486383334929, 8.816981222403210212250512039106, 9.400841679566366039215493598883, 10.36331169183046435625152979529, 10.81365970397423608373755426679, 11.18129072154113067986754908535, 11.61053766197120616313103490582, 11.88835318663780390861657362668, 12.31913464371876665447767818138