| L(s) = 1 | + 24·13-s − 24·19-s + 68·25-s + 152·31-s − 128·37-s − 80·43-s + 14·49-s − 48·61-s + 24·67-s + 16·73-s + 88·79-s + 160·97-s − 440·103-s − 176·109-s + 272·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 92·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 1.84·13-s − 1.26·19-s + 2.71·25-s + 4.90·31-s − 3.45·37-s − 1.86·43-s + 2/7·49-s − 0.786·61-s + 0.358·67-s + 0.219·73-s + 1.11·79-s + 1.64·97-s − 4.27·103-s − 1.61·109-s + 2.24·121-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 − 0.544·169-s + 0.00578·173-s + 0.00558·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(7.083596482\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.083596482\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 2294 T^{4} - 68 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 272 T^{2} + 36578 T^{4} - 272 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 12 T + 262 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 4 T^{2} - 59754 T^{4} - 4 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 12 T + 58 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 784 T^{2} + 386754 T^{4} - 784 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 1136 T^{2} + 1500194 T^{4} - 1136 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 76 T + 3338 T^{2} - 76 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 64 T + 3650 T^{2} + 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 2660 T^{2} + 5130134 T^{4} - 2660 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 40 T + 3090 T^{2} + 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 7252 T^{2} + 22326630 T^{4} - 7252 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 7168 T^{2} + 28480866 T^{4} - 7168 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 7732 T^{2} + 33955206 T^{4} - 7732 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 24 T - 2522 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 12 T + 3526 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 14800 T^{2} + 101192514 T^{4} - 14800 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 8 T + 9302 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )^{4} \) |
| 83 | $D_4\times C_2$ | \( 1 - 26404 T^{2} + 269064294 T^{4} - 26404 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 14084 T^{2} + 163244246 T^{4} - 14084 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 80 T + 20390 T^{2} - 80 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−6.82753125605395264293827103511, −6.76782633757088581149177939078, −6.54015573865219796197443838701, −6.27459742526365807765098895960, −6.09852221825006382643130843340, −5.99194383720089222280889228460, −5.47443214851210211555782185700, −5.29031259231644743768664186478, −5.10846985745362443831557457708, −4.79533844233064028416943745991, −4.63392597094170183341891274043, −4.34572503204871134625135751154, −4.21662635230860295570470495327, −3.89168620111824137921963709722, −3.34046681368507292961770438310, −3.33193088080871465899620881577, −3.15178201416089665066390568543, −2.72853192057293909084804595461, −2.62783404514569421704021010513, −2.05720592061107714346692608764, −1.61378974752904735096490992366, −1.60137362254515131294932137246, −1.05474174890426694192742606654, −0.62974104681610449471974648052, −0.51102602702475687564467860530,
0.51102602702475687564467860530, 0.62974104681610449471974648052, 1.05474174890426694192742606654, 1.60137362254515131294932137246, 1.61378974752904735096490992366, 2.05720592061107714346692608764, 2.62783404514569421704021010513, 2.72853192057293909084804595461, 3.15178201416089665066390568543, 3.33193088080871465899620881577, 3.34046681368507292961770438310, 3.89168620111824137921963709722, 4.21662635230860295570470495327, 4.34572503204871134625135751154, 4.63392597094170183341891274043, 4.79533844233064028416943745991, 5.10846985745362443831557457708, 5.29031259231644743768664186478, 5.47443214851210211555782185700, 5.99194383720089222280889228460, 6.09852221825006382643130843340, 6.27459742526365807765098895960, 6.54015573865219796197443838701, 6.76782633757088581149177939078, 6.82753125605395264293827103511
