| L(s) = 1 | − 2·4-s − 2·9-s + 6·11-s + 3·16-s − 14·19-s − 12·29-s − 4·31-s + 4·36-s − 6·41-s − 12·44-s + 10·49-s − 24·59-s − 4·61-s − 4·64-s − 2·71-s + 28·76-s − 26·79-s + 3·81-s − 50·89-s − 12·99-s + 6·101-s + 24·116-s + 15·121-s + 8·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 4-s − 2/3·9-s + 1.80·11-s + 3/4·16-s − 3.21·19-s − 2.22·29-s − 0.718·31-s + 2/3·36-s − 0.937·41-s − 1.80·44-s + 10/7·49-s − 3.12·59-s − 0.512·61-s − 1/2·64-s − 0.237·71-s + 3.21·76-s − 2.92·79-s + 1/3·81-s − 5.29·89-s − 1.20·99-s + 0.597·101-s + 2.22·116-s + 1.36·121-s + 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2930191129\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2930191129\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 7 | $C_2^2$ | \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $D_{4}$ | \( ( 1 - 3 T + 6 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 15 T^{2} + 376 T^{4} - 15 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 7 T^{2} - 304 T^{4} - 7 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 7 T + 32 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 31 | $D_{4}$ | \( ( 1 + 2 T - 10 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 99 T^{2} + 4732 T^{4} - 99 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 3 T + 66 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 111 T^{2} + 5884 T^{4} - 111 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 111 T^{2} + 6020 T^{4} - 111 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 61 | $D_{4}$ | \( ( 1 + 2 T + 50 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 120 T^{2} + 12286 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + T + 124 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 138 T^{2} + 14251 T^{4} - 138 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 13 T + 182 T^{2} + 13 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 165 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 25 T + 316 T^{2} + 25 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 80 T^{2} - 3234 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.01726224873405442689463826884, −5.94633900042620774772133921625, −5.62641437973552570817766773703, −5.58703683027526901376214404631, −5.44576166782236029087660637607, −5.03203241281339038326979409966, −4.71209872952103865229188774091, −4.58822106379877860456208792053, −4.50112493664960132739156434598, −4.10933760273608695210061244762, −3.99311842864866784368020487581, −3.88281240026503668013453134726, −3.86543782610498259709703700050, −3.53036774613882834292210671603, −3.02417666737965831720284879117, −2.87995449836068819897366693085, −2.77589361856562943826472469057, −2.54634401458542616601587730565, −1.95647343829635660082877364090, −1.74309218606890999731437430813, −1.63564762629728883461267356740, −1.51967853135372507620019725834, −1.05895620378190749750238118381, −0.36153152473413491898793648295, −0.14879498556167005692433561203,
0.14879498556167005692433561203, 0.36153152473413491898793648295, 1.05895620378190749750238118381, 1.51967853135372507620019725834, 1.63564762629728883461267356740, 1.74309218606890999731437430813, 1.95647343829635660082877364090, 2.54634401458542616601587730565, 2.77589361856562943826472469057, 2.87995449836068819897366693085, 3.02417666737965831720284879117, 3.53036774613882834292210671603, 3.86543782610498259709703700050, 3.88281240026503668013453134726, 3.99311842864866784368020487581, 4.10933760273608695210061244762, 4.50112493664960132739156434598, 4.58822106379877860456208792053, 4.71209872952103865229188774091, 5.03203241281339038326979409966, 5.44576166782236029087660637607, 5.58703683027526901376214404631, 5.62641437973552570817766773703, 5.94633900042620774772133921625, 6.01726224873405442689463826884
