| L(s) = 1 | − 16·9-s − 48·19-s − 18·25-s + 128·31-s + 128·49-s − 64·61-s + 288·79-s + 175·81-s + 320·109-s − 60·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 676·169-s + 768·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | − 1.77·9-s − 2.52·19-s − 0.719·25-s + 4.12·31-s + 2.61·49-s − 1.04·61-s + 3.64·79-s + 2.16·81-s + 2.93·109-s − 0.495·121-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 + 4·169-s + 4.49·171-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.958471537\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.958471537\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 + 16 T^{2} + p^{4} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 18 T^{2} + p^{4} T^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 - 64 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 + 450 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 12 T + p^{2} T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 480 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 31 | $C_2$ | \( ( 1 - 32 T + p^{2} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 2194 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 2032 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 3168 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 1010 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 6690 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 16 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 8944 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + 2942 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 72 T + p^{2} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 11856 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 11490 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 7838 T^{2} + p^{4} T^{4} )^{2} \) |
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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
−8.638606284034232253706828129191, −8.372593093865129606442053012032, −8.153756405755506133009992062690, −7.86837124194874197522108506312, −7.69838798651978055069919606691, −7.42071027170928327873516197194, −6.92479132313047137263418070103, −6.41957645686133779322421178328, −6.38157870445679999196167852734, −6.31568962816539996506819106434, −6.12437207313755619327312231213, −5.46442554630693547659182853113, −5.45397856280669469323429432632, −4.98427159931713622258385615574, −4.58988458551157477925149160282, −4.43434626288381453209627210693, −3.99806944319799776051741960806, −3.79214646943293589926134454599, −3.20168712648135074503162344981, −2.85606776514704660148920040694, −2.51930990812172090166982949046, −2.27096600072480738838875920779, −1.81723998877916494040463619015, −0.808008706225910223008799153076, −0.49165751528149475861300559833,
0.49165751528149475861300559833, 0.808008706225910223008799153076, 1.81723998877916494040463619015, 2.27096600072480738838875920779, 2.51930990812172090166982949046, 2.85606776514704660148920040694, 3.20168712648135074503162344981, 3.79214646943293589926134454599, 3.99806944319799776051741960806, 4.43434626288381453209627210693, 4.58988458551157477925149160282, 4.98427159931713622258385615574, 5.45397856280669469323429432632, 5.46442554630693547659182853113, 6.12437207313755619327312231213, 6.31568962816539996506819106434, 6.38157870445679999196167852734, 6.41957645686133779322421178328, 6.92479132313047137263418070103, 7.42071027170928327873516197194, 7.69838798651978055069919606691, 7.86837124194874197522108506312, 8.153756405755506133009992062690, 8.372593093865129606442053012032, 8.638606284034232253706828129191
