| L(s) = 1 | + 5·4-s − 8·5-s − 20·7-s − 5·9-s − 40·11-s + 16·16-s − 30·17-s − 40·20-s − 70·23-s + 66·25-s − 100·28-s + 160·35-s − 25·36-s + 40·43-s − 200·44-s + 40·45-s − 20·47-s + 54·49-s + 320·55-s + 80·61-s + 100·63-s + 115·64-s − 150·68-s − 210·73-s + 800·77-s − 128·80-s + 81·81-s + ⋯ |
| L(s) = 1 | + 5/4·4-s − 8/5·5-s − 2.85·7-s − 5/9·9-s − 3.63·11-s + 16-s − 1.76·17-s − 2·20-s − 3.04·23-s + 2.63·25-s − 3.57·28-s + 32/7·35-s − 0.694·36-s + 0.930·43-s − 4.54·44-s + 8/9·45-s − 0.425·47-s + 1.10·49-s + 5.81·55-s + 1.31·61-s + 1.58·63-s + 1.79·64-s − 2.20·68-s − 2.87·73-s + 10.3·77-s − 8/5·80-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.01229555833\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01229555833\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 19 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 - 5 T^{2} + 9 T^{4} - 5 p^{4} T^{6} + p^{8} T^{8} \) |
| 3 | $C_2^3$ | \( 1 + 5 T^{2} - 56 T^{4} + 5 p^{4} T^{6} + p^{8} T^{8} \) |
| 5 | $C_2^2$ | \( ( 1 + 4 T - 9 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 5 T + p^{2} T^{2} )^{4} \) |
| 11 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{4} \) |
| 13 | $C_2^3$ | \( 1 + 25 p T^{2} + 456 p^{2} T^{4} + 25 p^{5} T^{6} + p^{8} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 15 T - 64 T^{2} + 15 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 35 T + 696 T^{2} + 35 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^3$ | \( 1 + 1357 T^{2} + 1134168 T^{4} + 1357 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 - 622 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 2270 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^3$ | \( 1 + 2062 T^{2} + 1426083 T^{4} + 2062 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - 20 T - 1449 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 10 T - 2109 T^{2} + 10 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^3$ | \( 1 - 115 T^{2} - 7877256 T^{4} - 115 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $C_2^3$ | \( 1 + 6637 T^{2} + 31932408 T^{4} + 6637 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 40 T - 2121 T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 7405 T^{2} + 34682904 T^{4} + 7405 p^{4} T^{6} + p^{8} T^{8} \) |
| 71 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 92 T + 3423 T^{2} - 92 p^{2} T^{3} + p^{4} T^{4} )( 1 + 92 T + 3423 T^{2} + 92 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 73 | $C_2^2$ | \( ( 1 + 105 T + 5696 T^{2} + 105 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^3$ | \( 1 + 11182 T^{2} + 86087043 T^{4} + 11182 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $C_2$ | \( ( 1 + 40 T + p^{2} T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2^3$ | \( 1 + 3790 T^{2} - 74165181 T^{4} + 3790 p^{4} T^{6} + p^{8} T^{8} \) |
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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
−8.074420078630580475120229542661, −7.65869932553058423402139034523, −7.58647283508063833917452313082, −7.32349489720897859504077944525, −7.24401821741758027850053508175, −6.72267184123307630406626914669, −6.42644547196569178751452605669, −6.37402327418897642770215754102, −6.15257501643265000831503872658, −5.99081667550147139629672663850, −5.45231696645507253538154547248, −5.10293803599278816223949499799, −5.07670872235936993852582676363, −4.69011014358483051653908703082, −4.00946891412706680149392213966, −3.86188322478277541559634573788, −3.82319937266820744334255447300, −3.07803403817463508822691113928, −3.01323888623790190170852317818, −2.70736317452711992804966813205, −2.45567404469774321006084564884, −2.32474036861813042673378561035, −1.52771463952526017630268212589, −0.33168053368596443770838699718, −0.06091040918147189855721350513,
0.06091040918147189855721350513, 0.33168053368596443770838699718, 1.52771463952526017630268212589, 2.32474036861813042673378561035, 2.45567404469774321006084564884, 2.70736317452711992804966813205, 3.01323888623790190170852317818, 3.07803403817463508822691113928, 3.82319937266820744334255447300, 3.86188322478277541559634573788, 4.00946891412706680149392213966, 4.69011014358483051653908703082, 5.07670872235936993852582676363, 5.10293803599278816223949499799, 5.45231696645507253538154547248, 5.99081667550147139629672663850, 6.15257501643265000831503872658, 6.37402327418897642770215754102, 6.42644547196569178751452605669, 6.72267184123307630406626914669, 7.24401821741758027850053508175, 7.32349489720897859504077944525, 7.58647283508063833917452313082, 7.65869932553058423402139034523, 8.074420078630580475120229542661
