| L(s) = 1 | + 8·5-s + 24·13-s + 8·17-s + 4·25-s + 136·29-s − 40·37-s − 8·41-s + 100·49-s − 88·53-s − 296·61-s + 192·65-s + 88·73-s + 64·85-s − 216·89-s − 328·97-s − 24·101-s + 440·109-s + 312·113-s + 140·121-s − 72·125-s + 127-s + 131-s + 137-s + 139-s + 1.08e3·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 8/5·5-s + 1.84·13-s + 8/17·17-s + 4/25·25-s + 4.68·29-s − 1.08·37-s − 0.195·41-s + 2.04·49-s − 1.66·53-s − 4.85·61-s + 2.95·65-s + 1.20·73-s + 0.752·85-s − 2.42·89-s − 3.38·97-s − 0.237·101-s + 4.03·109-s + 2.76·113-s + 1.15·121-s − 0.575·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 7.50·145-s + 0.00671·149-s + 0.00662·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(4.243985653\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.243985653\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( ( 1 - 4 T + 22 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 5254 T^{4} - 100 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 5382 T^{4} - 140 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 12 T + 342 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 - 4 T + 454 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 780 T^{2} + 402374 T^{4} - 780 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 484 T^{2} + 517894 T^{4} - 484 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 68 T + 2806 T^{2} - 68 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 126 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 20 T + 1270 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 4 T + 2854 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 2764 T^{2} + 8710534 T^{4} - 2764 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 7428 T^{2} + 23258246 T^{4} - 7428 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 44 T + 6070 T^{2} + 44 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 11020 T^{2} + 52720774 T^{4} - 11020 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 148 T + 11350 T^{2} + 148 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 11980 T^{2} + 75342534 T^{4} - 11980 p^{4} T^{6} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 18276 T^{2} + 133865606 T^{4} - 18276 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 44 T + 9990 T^{2} - 44 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 1028 T^{2} + 28324230 T^{4} - 1028 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 5452 T^{2} + 97966918 T^{4} - 5452 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 108 T + 15558 T^{2} + 108 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 164 T + 22342 T^{2} + 164 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−6.81125283867734505342876355094, −6.61305073222305456755872921900, −6.22625227185466288804829429416, −6.05351852560833262930926439572, −5.88500301737042743491089433190, −5.88426142744681422413003726991, −5.55108766592641729578118744814, −5.44844050455904940744911690523, −4.74558532804764279152745288850, −4.71867040930663527999360604692, −4.57283466723973742930082598553, −4.54747047727654709425165837113, −4.02866305823115260208026311973, −3.72953669468041300617525627341, −3.21816053225964689935714969684, −3.20836054977621608618266975002, −3.12690653441728753669752836101, −2.71064459419176726937569400257, −2.22552855246060675926367930578, −2.14941232133283821755671965146, −1.74243779977826900137093323558, −1.26569773991839882038636133384, −1.08885056622549555231427795650, −1.06390857272983656068718462499, −0.23695901959675035100910703579,
0.23695901959675035100910703579, 1.06390857272983656068718462499, 1.08885056622549555231427795650, 1.26569773991839882038636133384, 1.74243779977826900137093323558, 2.14941232133283821755671965146, 2.22552855246060675926367930578, 2.71064459419176726937569400257, 3.12690653441728753669752836101, 3.20836054977621608618266975002, 3.21816053225964689935714969684, 3.72953669468041300617525627341, 4.02866305823115260208026311973, 4.54747047727654709425165837113, 4.57283466723973742930082598553, 4.71867040930663527999360604692, 4.74558532804764279152745288850, 5.44844050455904940744911690523, 5.55108766592641729578118744814, 5.88426142744681422413003726991, 5.88500301737042743491089433190, 6.05351852560833262930926439572, 6.22625227185466288804829429416, 6.61305073222305456755872921900, 6.81125283867734505342876355094
