| L(s) = 1 | − 20·5-s + 20·13-s − 32·17-s + 200·25-s + 140·29-s + 52·37-s − 100·49-s + 52·53-s − 236·61-s − 400·65-s + 126·81-s + 640·85-s + 480·97-s − 524·101-s + 76·109-s − 312·113-s − 1.50e3·125-s + 127-s + 131-s + 137-s + 139-s − 2.80e3·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 4·5-s + 1.53·13-s − 1.88·17-s + 8·25-s + 4.82·29-s + 1.40·37-s − 2.04·49-s + 0.981·53-s − 3.86·61-s − 6.15·65-s + 14/9·81-s + 7.52·85-s + 4.94·97-s − 5.18·101-s + 0.697·109-s − 2.76·113-s − 12·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 19.3·145-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-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
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.09844432989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09844432989\) |
| \(L(2)\) |
|
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 | $C_2^3$ | \( 1 - 14 p^{2} T^{4} + p^{8} T^{8} \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + p T )^{4}( 1 + p^{2} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 - 28606 T^{4} + p^{8} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - 10 T + 50 T^{2} - 10 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 8 T + p^{2} T^{2} )^{4} \) |
| 19 | $C_2^3$ | \( 1 - 54526 T^{4} + p^{8} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 + 1010 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 70 T + 2450 T^{2} - 70 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 1730 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 26 T + 338 T^{2} - 26 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 3106 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 + 2760002 T^{4} + p^{8} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + 382 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 26 T + 338 T^{2} - 26 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 + 23901122 T^{4} + p^{8} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 118 T + 6962 T^{2} + 118 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 9174914 T^{4} + p^{8} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 8882 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 7294 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 9410 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 23327234 T^{4} + p^{8} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 15806 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 120 T + p^{2} T^{2} )^{4} \) |
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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
−7.78481567815677173974575411290, −7.72670571081941112275324123682, −7.11689552290534702821898878111, −6.99036421862169736083725568282, −6.64403372055870286906016365103, −6.55768742639331659730134468699, −6.29268395716660769796691764375, −6.15695355598949598189074928247, −5.82455557653924991484631309472, −5.12162380603363668895811227797, −4.87117263833612410482752135073, −4.66983849068572709844484697373, −4.59602907759185143691287055176, −4.29988191823269818890349038038, −4.06633565707666797441922929685, −3.79544500450656063202023298977, −3.55223919062086724665091940357, −3.11581531896939131312770429181, −3.09849928790748595553041582579, −2.50091779567055000638758896584, −2.44525184572099052224724738791, −1.31586675446471643967621488493, −1.21617170123376658501463202603, −0.68902813468178228444572610765, −0.091293792872225584732489122931,
0.091293792872225584732489122931, 0.68902813468178228444572610765, 1.21617170123376658501463202603, 1.31586675446471643967621488493, 2.44525184572099052224724738791, 2.50091779567055000638758896584, 3.09849928790748595553041582579, 3.11581531896939131312770429181, 3.55223919062086724665091940357, 3.79544500450656063202023298977, 4.06633565707666797441922929685, 4.29988191823269818890349038038, 4.59602907759185143691287055176, 4.66983849068572709844484697373, 4.87117263833612410482752135073, 5.12162380603363668895811227797, 5.82455557653924991484631309472, 6.15695355598949598189074928247, 6.29268395716660769796691764375, 6.55768742639331659730134468699, 6.64403372055870286906016365103, 6.99036421862169736083725568282, 7.11689552290534702821898878111, 7.72670571081941112275324123682, 7.78481567815677173974575411290
