| L(s) = 1 | − 16·4-s + 472·13-s + 192·16-s − 40·19-s + 452·25-s + 136·31-s − 4.19e3·37-s + 9.58e3·43-s + 686·49-s − 7.55e3·52-s + 4.52e3·61-s − 2.04e3·64-s − 1.94e3·67-s − 2.13e4·73-s + 640·76-s + 1.03e4·79-s − 4.48e4·97-s − 7.23e3·100-s − 2.74e4·103-s + 3.32e4·109-s + 5.30e4·121-s − 2.17e3·124-s + 127-s + 131-s + 137-s + 139-s + 6.70e4·148-s + ⋯ |
| L(s) = 1 | − 4-s + 2.79·13-s + 3/4·16-s − 0.110·19-s + 0.723·25-s + 0.141·31-s − 3.06·37-s + 5.18·43-s + 2/7·49-s − 2.79·52-s + 1.21·61-s − 1/2·64-s − 0.433·67-s − 4.00·73-s + 0.110·76-s + 1.65·79-s − 4.76·97-s − 0.723·100-s − 2.59·103-s + 2.79·109-s + 3.62·121-s − 0.141·124-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 3.06·148-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.223826447\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.223826447\) |
| \(L(3)\) |
|
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 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 452 T^{2} - 95146 T^{4} - 452 p^{8} T^{6} + p^{16} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 53072 T^{2} + 1132855106 T^{4} - 53072 p^{8} T^{6} + p^{16} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 236 T + 65558 T^{2} - 236 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 229220 T^{2} + 24556773974 T^{4} - 229220 p^{8} T^{6} + p^{16} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 20 T - 47958 T^{2} + 20 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 918512 T^{2} + 366465886466 T^{4} - 918512 p^{8} T^{6} + p^{16} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 2538896 T^{2} + 2606258922434 T^{4} - 2538896 p^{8} T^{6} + p^{16} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 68 T + 1846826 T^{2} - 68 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 2096 T + 3611826 T^{2} + 2096 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 1782916 T^{2} - 5802193925994 T^{4} - 1782916 p^{8} T^{6} + p^{16} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 4792 T + 12529026 T^{2} - 4792 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 9480916 T^{2} + 51658338912486 T^{4} - 9480916 p^{8} T^{6} + p^{16} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 7270304 T^{2} + 133516740466626 T^{4} + 7270304 p^{8} T^{6} + p^{16} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 22659476 T^{2} + 261789383129894 T^{4} - 22659476 p^{8} T^{6} + p^{16} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 2264 T + 28247318 T^{2} - 2264 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 972 T + 38557270 T^{2} + 972 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 + 33883024 T^{2} + 1543886951212034 T^{4} + 33883024 p^{8} T^{6} + p^{16} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 10680 T + 75860374 T^{2} + 10680 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 5156 T + 84195014 T^{2} - 5156 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 34049348 T^{2} + 133007755977158 T^{4} - 34049348 p^{8} T^{6} + p^{16} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 233538340 T^{2} + 21473360777771862 T^{4} - 233538340 p^{8} T^{6} + p^{16} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 22416 T + 291316294 T^{2} + 22416 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
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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
−9.050162166978576244646696840956, −8.733255191407748375185217367685, −8.457481781263099841229498629646, −8.444474134972974822392041405699, −8.345067105029174773329087348888, −7.49279057613902829355580114429, −7.38185458854658423558087371076, −7.19405674867366823572398968691, −6.84222540735412665366447575706, −6.20232472289525810763756875696, −6.04615944926057010483982869785, −5.82269142190973505008299965550, −5.66320314393594585771416730434, −4.99836062702803397095693000757, −4.90240684170773841963452363987, −4.25551464101046207186849567762, −3.99215145161930996402660448083, −3.81840336332721134647392495452, −3.48249062680711738531672500746, −2.84318922966505863485638512958, −2.58863598998658717784366717111, −1.76924532023850163274871726590, −1.15986493611242362157461090787, −1.09525492731343963434670055082, −0.33143879563410006776039891519,
0.33143879563410006776039891519, 1.09525492731343963434670055082, 1.15986493611242362157461090787, 1.76924532023850163274871726590, 2.58863598998658717784366717111, 2.84318922966505863485638512958, 3.48249062680711738531672500746, 3.81840336332721134647392495452, 3.99215145161930996402660448083, 4.25551464101046207186849567762, 4.90240684170773841963452363987, 4.99836062702803397095693000757, 5.66320314393594585771416730434, 5.82269142190973505008299965550, 6.04615944926057010483982869785, 6.20232472289525810763756875696, 6.84222540735412665366447575706, 7.19405674867366823572398968691, 7.38185458854658423558087371076, 7.49279057613902829355580114429, 8.345067105029174773329087348888, 8.444474134972974822392041405699, 8.457481781263099841229498629646, 8.733255191407748375185217367685, 9.050162166978576244646696840956
