| L(s) = 1 | + 72·17-s − 76·25-s − 216·41-s + 188·49-s − 40·73-s − 72·89-s − 136·97-s − 360·113-s + 100·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 188·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 4.23·17-s − 3.03·25-s − 5.26·41-s + 3.83·49-s − 0.547·73-s − 0.808·89-s − 1.40·97-s − 3.18·113-s + 0.826·121-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 + 1.11·169-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 + 0.00502·199-s + 0.00473·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.281375940\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.281375940\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 18 T + p^{2} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 290 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 238 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 710 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 1438 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 1010 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 54 T + p^{2} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 3266 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 3122 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 4474 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 3074 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 5090 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 1582 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 9982 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 13586 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 18 T + p^{2} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 34 T + p^{2} T^{2} )^{4} \) |
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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
−6.07045181940258711677271354676, −5.79808770915859475070445701763, −5.78075041926975994405866492294, −5.60814377922112233866507038295, −5.32972786236860816435231890390, −5.27466707646920596281110672966, −5.24181586063049763988165489289, −4.64197612385890081912801446061, −4.63502026540535267666422667269, −4.07589132168543750839261375208, −4.03179492751908042416226767094, −3.81912763757014927954831060414, −3.60030202344744116065612202545, −3.37499114718990551931288490770, −3.26081119285340019929505601886, −2.87196078648473092238224030275, −2.76178910322190003686337738467, −2.39489635718795690160518039094, −1.96081875340444630844393176502, −1.70325571006704654315042559556, −1.54133065481607711135447190067, −1.39009292994680560335532760675, −0.865188031110148800678336184681, −0.58256539221605729746424592267, −0.23471710417514245836416505578,
0.23471710417514245836416505578, 0.58256539221605729746424592267, 0.865188031110148800678336184681, 1.39009292994680560335532760675, 1.54133065481607711135447190067, 1.70325571006704654315042559556, 1.96081875340444630844393176502, 2.39489635718795690160518039094, 2.76178910322190003686337738467, 2.87196078648473092238224030275, 3.26081119285340019929505601886, 3.37499114718990551931288490770, 3.60030202344744116065612202545, 3.81912763757014927954831060414, 4.03179492751908042416226767094, 4.07589132168543750839261375208, 4.63502026540535267666422667269, 4.64197612385890081912801446061, 5.24181586063049763988165489289, 5.27466707646920596281110672966, 5.32972786236860816435231890390, 5.60814377922112233866507038295, 5.78075041926975994405866492294, 5.79808770915859475070445701763, 6.07045181940258711677271354676
