| L(s) = 1 | − 2·2-s + 2·4-s − 4·5-s + 11·9-s + 8·10-s − 20·13-s + 10·17-s − 22·18-s − 8·20-s − 26·25-s + 40·26-s − 30·29-s − 8·32-s − 20·34-s + 22·36-s + 16·41-s − 44·45-s − 25·49-s + 52·50-s − 40·52-s − 30·53-s + 60·58-s + 16·64-s + 80·65-s + 20·68-s − 10·73-s + 69·81-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 1.78·5-s + 11/3·9-s + 2.52·10-s − 5.54·13-s + 2.42·17-s − 5.18·18-s − 1.78·20-s − 5.19·25-s + 7.84·26-s − 5.57·29-s − 1.41·32-s − 3.42·34-s + 11/3·36-s + 2.49·41-s − 6.55·45-s − 3.57·49-s + 7.35·50-s − 5.54·52-s − 4.12·53-s + 7.87·58-s + 2·64-s + 9.92·65-s + 2.42·68-s − 1.17·73-s + 23/3·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.03394426974\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03394426974\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T + p T^{2} - p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{7} + p^{4} T^{8} \) |
| 31 | \( 1 + 79 T^{2} + 3381 T^{4} + 79 p^{2} T^{6} + p^{4} T^{8} \) |
| good | 3 | \( 1 - 11 T^{2} + 52 T^{4} - 17 p^{2} T^{6} + 415 T^{8} - 17 p^{4} T^{10} + 52 p^{4} T^{12} - 11 p^{6} T^{14} + p^{8} T^{16} \) |
| 5 | \( ( 1 + T + 9 T^{2} + p T^{3} + p^{2} T^{4} )^{4} \) |
| 7 | \( 1 + 25 T^{2} + 216 T^{4} + 335 T^{6} - 4129 T^{8} + 335 p^{2} T^{10} + 216 p^{4} T^{12} + 25 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( ( 1 + 19 T^{2} + 181 T^{4} + 19 p^{2} T^{6} + p^{4} T^{8} )( 1 + 19 T^{2} + 301 T^{4} + 19 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 13 | \( ( 1 + 10 T + 43 T^{2} + 80 T^{3} + 109 T^{4} + 80 p T^{5} + 43 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 5 T + 42 T^{2} - 215 T^{3} + 839 T^{4} - 215 p T^{5} + 42 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 19 | \( 1 + 13 T^{2} - 192 T^{4} - 7189 T^{6} - 24145 T^{8} - 7189 p^{2} T^{10} - 192 p^{4} T^{12} + 13 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( 1 - 51 T^{2} + 1212 T^{4} - 8113 T^{6} - 52185 T^{8} - 8113 p^{2} T^{10} + 1212 p^{4} T^{12} - 51 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( ( 1 + 15 T + 149 T^{2} + 1165 T^{3} + 6956 T^{4} + 1165 p T^{5} + 149 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 58 T^{2} + 1959 T^{4} - 58 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 8 T - 17 T^{2} + 254 T^{3} - 435 T^{4} + 254 p T^{5} - 17 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 - 61 T^{2} + 3647 T^{4} - 4421 p T^{6} + 5755480 T^{8} - 4421 p^{3} T^{10} + 3647 p^{4} T^{12} - 61 p^{6} T^{14} + p^{8} T^{16} \) |
| 47 | \( 1 + 95 T^{2} + 3351 T^{4} - 62555 T^{6} - 10362664 T^{8} - 62555 p^{2} T^{10} + 3351 p^{4} T^{12} + 95 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( ( 1 + 15 T + 98 T^{2} - 135 T^{3} - 3941 T^{4} - 135 p T^{5} + 98 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 + 117 T^{2} + 6168 T^{4} + 43019 T^{6} - 12338385 T^{8} + 43019 p^{2} T^{10} + 6168 p^{4} T^{12} + 117 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( ( 1 - 144 T^{2} + 10206 T^{4} - 144 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 130 T^{2} + 11923 T^{4} - 130 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( 1 + 83 T^{2} + 7623 T^{4} + 550081 T^{6} + 17574680 T^{8} + 550081 p^{2} T^{10} + 7623 p^{4} T^{12} + 83 p^{6} T^{14} + p^{8} T^{16} \) |
| 73 | \( ( 1 + 5 T + 83 T^{2} + 375 T^{3} + 6064 T^{4} + 375 p T^{5} + 83 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( 1 - 218 T^{2} + 16203 T^{4} - 693136 T^{6} + 44349125 T^{8} - 693136 p^{2} T^{10} + 16203 p^{4} T^{12} - 218 p^{6} T^{14} + p^{8} T^{16} \) |
| 83 | \( 1 - 126 T^{2} + 8367 T^{4} - 143788 T^{6} - 39412275 T^{8} - 143788 p^{2} T^{10} + 8367 p^{4} T^{12} - 126 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( ( 1 + 25 T + 489 T^{2} + 6415 T^{3} + 69356 T^{4} + 6415 p T^{5} + 489 p^{2} T^{6} + 25 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 13 T - 28 T^{2} + 505 T^{3} + 3031 T^{4} + 505 p T^{5} - 28 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−6.26383564222240616521385959331, −6.19765089815872044207430549317, −5.92782686234689982943791875603, −5.72935621483113726394901771416, −5.52008176191281440536881241188, −5.46781500884154482555163403809, −5.43440821100125345400309393070, −4.87841020033375319656909219101, −4.78447691672486097476754939985, −4.67689258372987025437537345968, −4.45077137425536715058083495886, −4.41988556621162438659809875054, −4.24965379250824739476399692725, −3.84661500504135174801916353175, −3.70293624395233719707274430253, −3.66199326770381031547962901441, −3.44766601088981905020414298044, −3.12701279245170466015009988611, −3.02168084891287436226752979613, −2.14835583579567406584730376129, −2.02530258016256654434371266501, −1.96446641156117055601860268381, −1.76557261619167372953523723848, −1.66045752618035996595193436747, −0.15855955388960393802447133357,
0.15855955388960393802447133357, 1.66045752618035996595193436747, 1.76557261619167372953523723848, 1.96446641156117055601860268381, 2.02530258016256654434371266501, 2.14835583579567406584730376129, 3.02168084891287436226752979613, 3.12701279245170466015009988611, 3.44766601088981905020414298044, 3.66199326770381031547962901441, 3.70293624395233719707274430253, 3.84661500504135174801916353175, 4.24965379250824739476399692725, 4.41988556621162438659809875054, 4.45077137425536715058083495886, 4.67689258372987025437537345968, 4.78447691672486097476754939985, 4.87841020033375319656909219101, 5.43440821100125345400309393070, 5.46781500884154482555163403809, 5.52008176191281440536881241188, 5.72935621483113726394901771416, 5.92782686234689982943791875603, 6.19765089815872044207430549317, 6.26383564222240616521385959331
Plot not available for L-functions of degree greater than 10.
