L(s) = 1 | − 4·3-s + 8·9-s − 8·13-s − 28·25-s − 20·27-s − 48·31-s + 32·39-s + 48·49-s − 40·73-s + 112·75-s + 50·81-s + 192·93-s − 64·117-s − 28·121-s + 127-s + 131-s + 137-s + 139-s − 192·147-s + 149-s + 151-s + 157-s + 163-s + 167-s − 48·169-s + 173-s + 179-s + ⋯ |
L(s) = 1 | − 2.30·3-s + 8/3·9-s − 2.21·13-s − 5.59·25-s − 3.84·27-s − 8.62·31-s + 5.12·39-s + 48/7·49-s − 4.68·73-s + 12.9·75-s + 50/9·81-s + 19.9·93-s − 5.91·117-s − 2.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 15.8·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 3.69·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2486687741\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2486687741\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
good | 5 | \( ( 1 + 14 T^{2} + 94 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 - 12 T^{2} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 + 14 T^{2} + 166 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 - 16 T^{2} + 322 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 6 T^{2} - 394 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 88 T^{2} + 3438 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 6 T + p T^{2} )^{8} \) |
| 37 | \( ( 1 + 26 T^{2} + 2062 T^{4} + 26 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 92 T^{2} + 4198 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 38 T^{2} + 2934 T^{4} - 38 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 48 T^{2} + 494 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + 206 T^{2} + 16222 T^{4} + 206 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 174 T^{2} + 13886 T^{4} - 174 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 214 T^{2} + 20182 T^{4} - 214 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 32 T^{2} + 9838 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 10 T + 166 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 120 T^{2} + 13202 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 246 T^{2} + 28502 T^{4} + 246 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 320 T^{2} + 41122 T^{4} + 320 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 108 T^{2} + 3734 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−4.68119015946928778833445014550, −4.52927282664644909688462798321, −4.47818695718296167966229949562, −4.43329276268403949784208836061, −4.10580493295389063352061380055, −3.98369827803966235198981914841, −3.96538771151568190099989655306, −3.74188980756035717074216153232, −3.66554235226217017543402643431, −3.57881652777935217480998624550, −3.48562584613584815460545328866, −3.23988662914998256573579790462, −3.03413336912760846863473329961, −2.47288230145750510600577473624, −2.45928763435267401814922324179, −2.43448093473988784236797999493, −2.15564736879325964523126007044, −2.05940171217713392051953755433, −1.76448485705016061065564578608, −1.67504871494726200973639065443, −1.66714262347443366177864946724, −1.35672409607243476044484771335, −0.43322026802330067264163923141, −0.35858432026172426610123530756, −0.34409592766589972110314568277,
0.34409592766589972110314568277, 0.35858432026172426610123530756, 0.43322026802330067264163923141, 1.35672409607243476044484771335, 1.66714262347443366177864946724, 1.67504871494726200973639065443, 1.76448485705016061065564578608, 2.05940171217713392051953755433, 2.15564736879325964523126007044, 2.43448093473988784236797999493, 2.45928763435267401814922324179, 2.47288230145750510600577473624, 3.03413336912760846863473329961, 3.23988662914998256573579790462, 3.48562584613584815460545328866, 3.57881652777935217480998624550, 3.66554235226217017543402643431, 3.74188980756035717074216153232, 3.96538771151568190099989655306, 3.98369827803966235198981914841, 4.10580493295389063352061380055, 4.43329276268403949784208836061, 4.47818695718296167966229949562, 4.52927282664644909688462798321, 4.68119015946928778833445014550
Plot not available for L-functions of degree greater than 10.