| L(s) = 1 | − 24·11-s − 104·31-s − 120·41-s + 8·61-s − 480·71-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 + 229-s + ⋯ |
| L(s) = 1 | − 2.18·11-s − 3.35·31-s − 2.92·41-s + 8/61·61-s − 6.76·71-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 + 0.00436·229-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6128316294\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6128316294\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 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
−7.04850385462747315471939168706, −7.02178768411567765806345527940, −6.70579949622583208395959603893, −6.28749672020372424188099641581, −5.93945931113551269534745328449, −5.91465165038303376640931720317, −5.65015929070116312502542301791, −5.49864781097268323275071794062, −5.04620310732192368520969275723, −4.95410027623589559686490170891, −4.90278136500489287606369625058, −4.59117989420214623872995806565, −3.98763558350379619207588470451, −3.85988694484407953008263160258, −3.81705667252396018420030352085, −3.32141014826691469664054860195, −2.90357349290286583319614447958, −2.79776663509598601636133358458, −2.77191517678768151072186394101, −2.07333524329183594228778075136, −1.85043691558836268875502946036, −1.45927980659828016075305454362, −1.43104362738419503809396537452, −0.32711401710055557720219582306, −0.24540925716943471800413213010,
0.24540925716943471800413213010, 0.32711401710055557720219582306, 1.43104362738419503809396537452, 1.45927980659828016075305454362, 1.85043691558836268875502946036, 2.07333524329183594228778075136, 2.77191517678768151072186394101, 2.79776663509598601636133358458, 2.90357349290286583319614447958, 3.32141014826691469664054860195, 3.81705667252396018420030352085, 3.85988694484407953008263160258, 3.98763558350379619207588470451, 4.59117989420214623872995806565, 4.90278136500489287606369625058, 4.95410027623589559686490170891, 5.04620310732192368520969275723, 5.49864781097268323275071794062, 5.65015929070116312502542301791, 5.91465165038303376640931720317, 5.93945931113551269534745328449, 6.28749672020372424188099641581, 6.70579949622583208395959603893, 7.02178768411567765806345527940, 7.04850385462747315471939168706
