| L(s) = 1 | − 12·4-s + 76·16-s − 100·19-s − 156·31-s + 34·49-s + 292·61-s − 288·64-s + 1.20e3·76-s + 96·79-s + 100·109-s − 192·121-s + 1.87e3·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 674·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 3·4-s + 19/4·16-s − 5.26·19-s − 5.03·31-s + 0.693·49-s + 4.78·61-s − 9/2·64-s + 15.7·76-s + 1.21·79-s + 0.917·109-s − 1.58·121-s + 15.0·124-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 + 3.98·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.04449920103\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04449920103\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( ( 1 + 3 p T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 17 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 96 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 337 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 144 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 25 T + p^{2} T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 1040 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 960 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 39 T + p^{2} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 1714 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 3330 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 3169 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 3360 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 3630 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 6864 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 73 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 5009 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 6210 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 7838 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 24 T + p^{2} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 11600 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 146 T + p^{2} T^{2} )^{2}( 1 + 146 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 18769 T^{2} + p^{4} T^{4} )^{2} \) |
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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
−8.701898940307724008625165957281, −8.586475717743626918410703000928, −8.504353338432426933027634306887, −8.067998430976131239372835984718, −7.81161061384981092440207150973, −7.45107828840233738354518272851, −7.12274354305100634448279968914, −6.75792049500721193427429627186, −6.43723931929326329169989754538, −6.36870648191100473249354892716, −5.74713748635186946928795010304, −5.42706721921762772217013368174, −5.41126079327799684409229033925, −5.01628279795112691035567279412, −4.73716502826992248513158133671, −4.11453094233050258327389252582, −4.08260669534730046180297217563, −4.01401840891701747357018558550, −3.75909257093194343656265699079, −3.29010064785278608502816317874, −2.24414519363540639149870404666, −2.22808630141681842520291173513, −1.77966314376157158665249237127, −0.66203721288307329801288480067, −0.097613318755117268343618852871,
0.097613318755117268343618852871, 0.66203721288307329801288480067, 1.77966314376157158665249237127, 2.22808630141681842520291173513, 2.24414519363540639149870404666, 3.29010064785278608502816317874, 3.75909257093194343656265699079, 4.01401840891701747357018558550, 4.08260669534730046180297217563, 4.11453094233050258327389252582, 4.73716502826992248513158133671, 5.01628279795112691035567279412, 5.41126079327799684409229033925, 5.42706721921762772217013368174, 5.74713748635186946928795010304, 6.36870648191100473249354892716, 6.43723931929326329169989754538, 6.75792049500721193427429627186, 7.12274354305100634448279968914, 7.45107828840233738354518272851, 7.81161061384981092440207150973, 8.067998430976131239372835984718, 8.504353338432426933027634306887, 8.586475717743626918410703000928, 8.701898940307724008625165957281
