| L(s) = 1 | + 8·4-s − 52·7-s − 100·13-s − 1.43e3·19-s − 962·25-s − 416·28-s + 1.48e3·31-s + 7.49e3·37-s + 524·43-s + 5.47e3·49-s − 800·52-s + 2.97e3·61-s − 512·64-s + 8.97e3·67-s + 1.16e3·73-s − 1.14e4·76-s − 1.96e4·79-s + 5.20e3·91-s + 956·97-s − 7.69e3·100-s − 4.27e3·103-s − 1.90e4·109-s − 1.51e4·121-s + 1.18e4·124-s + 127-s + 131-s + 7.44e4·133-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 1.06·7-s − 0.591·13-s − 3.96·19-s − 1.53·25-s − 0.530·28-s + 1.54·31-s + 5.47·37-s + 0.283·43-s + 2.28·49-s − 0.295·52-s + 0.798·61-s − 1/8·64-s + 1.99·67-s + 0.217·73-s − 1.98·76-s − 3.14·79-s + 0.627·91-s + 0.101·97-s − 0.769·100-s − 0.403·103-s − 1.59·109-s − 1.03·121-s + 0.772·124-s + 6.20e−5·127-s + 5.82e−5·131-s + 4.20·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.4248493097\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4248493097\) |
| \(L(3)\) |
|
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 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^3$ | \( 1 + 962 T^{2} + 534819 T^{4} + 962 p^{8} T^{6} + p^{16} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 + 26 T - 1725 T^{2} + 26 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 + 15170 T^{2} + 15770019 T^{4} + 15170 p^{8} T^{6} + p^{16} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 50 T - 26061 T^{2} + 50 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 125570 T^{2} + p^{8} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 358 T + p^{4} T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 + 420290 T^{2} + 98332698819 T^{4} + 420290 p^{8} T^{6} + p^{16} T^{8} \) |
| 29 | $C_2^3$ | \( 1 - 666238 T^{2} - 56373340317 T^{4} - 666238 p^{8} T^{6} + p^{16} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 - 742 T - 372957 T^{2} - 742 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 1874 T + p^{4} T^{2} )^{4} \) |
| 41 | $C_2^3$ | \( 1 - 155710 T^{2} - 7960679625021 T^{4} - 155710 p^{8} T^{6} + p^{16} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - 262 T - 3350157 T^{2} - 262 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 6879362 T^{2} + 23514334865283 T^{4} + 6879362 p^{8} T^{6} + p^{16} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 15571010 T^{2} + p^{8} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 + 20937410 T^{2} + 291544699903779 T^{4} + 20937410 p^{8} T^{6} + p^{16} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 1486 T - 11637645 T^{2} - 1486 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 4486 T - 26925 T^{2} - 4486 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 38122562 T^{2} + p^{8} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 290 T + p^{4} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 9818 T + 57443043 T^{2} + 9818 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 44355074 T^{2} - 284919642593565 T^{4} + 44355074 p^{8} T^{6} + p^{16} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 64012610 T^{2} + p^{8} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 478 T - 88300797 T^{2} - 478 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
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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
−8.445502512821002622391400459250, −8.425154473466798516065946635365, −8.320270396806837551300163945172, −7.972434187511605704640464599994, −7.49205971872912849240911289282, −7.35134665601089930317182040174, −6.96908260316348561630259711577, −6.72070959724513484162987727972, −6.24166446451112551444127057543, −6.07337376885262980019623789023, −6.05177458414300702227362903841, −5.89950873471569240370643600700, −5.17961326965401122912651638149, −4.67722321763150413745803402558, −4.33979985623070319917160291002, −4.32238274303068033218317618698, −3.76493477226421774194291274904, −3.75296069944931287669167627195, −2.62680719890207985809393213174, −2.54327814729763840530801060169, −2.53972040498503361929803434268, −2.09934309160725662668226280035, −1.23807423117716835701704727028, −0.789973166643029950773825230337, −0.12992366597606757807257192283,
0.12992366597606757807257192283, 0.789973166643029950773825230337, 1.23807423117716835701704727028, 2.09934309160725662668226280035, 2.53972040498503361929803434268, 2.54327814729763840530801060169, 2.62680719890207985809393213174, 3.75296069944931287669167627195, 3.76493477226421774194291274904, 4.32238274303068033218317618698, 4.33979985623070319917160291002, 4.67722321763150413745803402558, 5.17961326965401122912651638149, 5.89950873471569240370643600700, 6.05177458414300702227362903841, 6.07337376885262980019623789023, 6.24166446451112551444127057543, 6.72070959724513484162987727972, 6.96908260316348561630259711577, 7.35134665601089930317182040174, 7.49205971872912849240911289282, 7.972434187511605704640464599994, 8.320270396806837551300163945172, 8.425154473466798516065946635365, 8.445502512821002622391400459250
