| L(s) = 1 | + 4·2-s − 4·3-s + 8·4-s − 16·6-s + 8·9-s − 32·12-s − 36·16-s + 4·17-s + 32·18-s + 12·23-s − 8·27-s + 4·31-s − 96·32-s + 16·34-s + 64·36-s + 48·46-s − 16·47-s + 144·48-s − 16·51-s − 8·53-s − 32·54-s − 16·61-s + 16·62-s − 96·64-s + 32·68-s − 48·69-s − 7·81-s + ⋯ |
| L(s) = 1 | + 2.82·2-s − 2.30·3-s + 4·4-s − 6.53·6-s + 8/3·9-s − 9.23·12-s − 9·16-s + 0.970·17-s + 7.54·18-s + 2.50·23-s − 1.53·27-s + 0.718·31-s − 16.9·32-s + 2.74·34-s + 32/3·36-s + 7.07·46-s − 2.33·47-s + 20.7·48-s − 2.24·51-s − 1.09·53-s − 4.35·54-s − 2.04·61-s + 2.03·62-s − 12·64-s + 3.88·68-s − 5.77·69-s − 7/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.436550755\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.436550755\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 4 T + 8 T^{2} + 8 T^{3} + 7 T^{4} + 8 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 23 T^{4} + p^{4} T^{8} \) |
| good | 2 | \( ( 1 - p T + p T^{2} )^{4}( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | \( ( 1 + 4 T^{2} - 105 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 337 T^{4} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 2 T + 2 T^{2} + 64 T^{3} - 353 T^{4} + 64 p T^{5} + 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{2}( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | \( ( 1 - 6 T + 18 T^{2} + 168 T^{3} - 1033 T^{4} + 168 p T^{5} + 18 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - T - 30 T^{2} - p T^{3} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - 2062 T^{4} + p^{4} T^{8} )( 1 - 529 T^{4} + p^{4} T^{8} ) \) |
| 41 | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 + 23 T^{4} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 8 T + 32 T^{2} - 496 T^{3} - 4193 T^{4} - 496 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + 4 T + 8 T^{2} - 392 T^{3} - 3593 T^{4} - 392 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 116 T^{2} + 9975 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 4 T - 45 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( 1 - 8711 T^{4} + 55730400 T^{8} - 8711 p^{4} T^{12} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 92 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 8542 T^{4} + p^{4} T^{8} )( 1 + 9791 T^{4} + p^{4} T^{8} ) \) |
| 79 | \( ( 1 + 109 T^{2} + 5640 T^{4} + 109 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 + 64 T^{2} - 3825 T^{4} + 64 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 24 T + 288 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{2}( 1 + 24 T + 288 T^{2} + 24 p T^{3} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.83319252247745026109728723870, −4.66423355066277632656087586523, −4.61396948030042215611991400713, −4.55897040463544383027393430408, −4.18709311932766216630213427482, −4.09166026862852091018255908299, −4.01247823140068807454590808698, −3.87899444017113932181020423798, −3.57807404747232886283128585675, −3.47533629721085880765523570330, −3.17097766642364988303708366285, −3.13096686300770899409732940220, −3.11443460388783436993011380845, −3.06294128445661924585292759233, −2.93429037672516484992005517379, −2.69173069825946956980170462098, −2.36569853500982379244478981913, −2.25331367513189857063684935753, −2.11071668443074213672614576032, −1.80453043002249412808243547797, −1.26898138976229541699222355016, −1.23198412888851821063801694677, −1.14042827167282646071970407426, −0.38184905351482971880910528668, −0.26701806996592273817600463919,
0.26701806996592273817600463919, 0.38184905351482971880910528668, 1.14042827167282646071970407426, 1.23198412888851821063801694677, 1.26898138976229541699222355016, 1.80453043002249412808243547797, 2.11071668443074213672614576032, 2.25331367513189857063684935753, 2.36569853500982379244478981913, 2.69173069825946956980170462098, 2.93429037672516484992005517379, 3.06294128445661924585292759233, 3.11443460388783436993011380845, 3.13096686300770899409732940220, 3.17097766642364988303708366285, 3.47533629721085880765523570330, 3.57807404747232886283128585675, 3.87899444017113932181020423798, 4.01247823140068807454590808698, 4.09166026862852091018255908299, 4.18709311932766216630213427482, 4.55897040463544383027393430408, 4.61396948030042215611991400713, 4.66423355066277632656087586523, 4.83319252247745026109728723870
Plot not available for L-functions of degree greater than 10.
