| L(s) = 1 | − 32·9-s − 64·17-s − 28·25-s − 72·41-s − 60·49-s − 320·73-s + 606·81-s − 64·89-s + 192·97-s + 632·113-s − 160·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 2.04e3·153-s + 157-s + 163-s + 167-s + 548·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 3.55·9-s − 3.76·17-s − 1.11·25-s − 1.75·41-s − 1.22·49-s − 4.38·73-s + 7.48·81-s − 0.719·89-s + 1.97·97-s + 5.59·113-s − 1.32·121-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 + 13.3·153-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 3.24·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.08876260526\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08876260526\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( ( 1 + 16 T^{2} + p^{4} T^{4} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 6 T + p^{2} T^{2} )^{2}( 1 + 6 T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 80 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 274 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 16 T + p^{2} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + 720 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 94 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 1618 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 1410 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 1138 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 18 T + p^{2} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 656 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 2370 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 2126 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 6800 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 5842 T^{2} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 3504 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 6882 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 80 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 7874 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 9360 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 16 T + p^{2} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 48 T + p^{2} T^{2} )^{4} \) |
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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
−7.57595155403976771641959229119, −7.36488450443621215337458727514, −7.26500185770520259123416112765, −6.89015460132426779674589752147, −6.49308142331529317596653062810, −6.31974713596185112772568885371, −6.19665805479626428373299445982, −6.08510397237921546950320908340, −5.68295940936271970927875117359, −5.41268658766431061289541407572, −5.27010821565512282707454583951, −4.86148750902609438192914443709, −4.51663256600629764384760577604, −4.48600360166447001143660435633, −4.12790859049862202457524225634, −3.64099283557891341605534829187, −3.25503597688476049380387227059, −3.02701939851977876146791152516, −3.01090894488742012310940527914, −2.23101356300062580831603811965, −2.21717444925389840487941028538, −2.07268840064930662499558077081, −1.38685938410818028536311888649, −0.41717973376242018310648691213, −0.10557602866914321786110059686,
0.10557602866914321786110059686, 0.41717973376242018310648691213, 1.38685938410818028536311888649, 2.07268840064930662499558077081, 2.21717444925389840487941028538, 2.23101356300062580831603811965, 3.01090894488742012310940527914, 3.02701939851977876146791152516, 3.25503597688476049380387227059, 3.64099283557891341605534829187, 4.12790859049862202457524225634, 4.48600360166447001143660435633, 4.51663256600629764384760577604, 4.86148750902609438192914443709, 5.27010821565512282707454583951, 5.41268658766431061289541407572, 5.68295940936271970927875117359, 6.08510397237921546950320908340, 6.19665805479626428373299445982, 6.31974713596185112772568885371, 6.49308142331529317596653062810, 6.89015460132426779674589752147, 7.26500185770520259123416112765, 7.36488450443621215337458727514, 7.57595155403976771641959229119
