| L(s) = 1 | + 8·9-s − 8·11-s − 16·23-s + 4·25-s − 16·37-s − 14·49-s − 16·53-s + 48·67-s − 32·71-s + 30·81-s − 64·99-s + 24·113-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 24·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 8/3·9-s − 2.41·11-s − 3.33·23-s + 4/5·25-s − 2.63·37-s − 2·49-s − 2.19·53-s + 5.86·67-s − 3.79·71-s + 10/3·81-s − 6.43·99-s + 2.25·113-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.84·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1387950910\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1387950910\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| good | 3 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 12 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 60 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + 68 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 - 116 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 162 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.88318775903522271829812548419, −6.80965300056167292998699194698, −6.76612360278980203045233818831, −6.23408767164125756187837618587, −6.00630590992215105568389779806, −5.88215485265376661954309589454, −5.67682839751901155424779844417, −5.22240350698829095250257634884, −5.09340614129438693617138393357, −4.87359054721853402831687773341, −4.84860549579251025626369382622, −4.33349365851407419642772930446, −4.18133568967525758006925552893, −4.14747976697602489594439922433, −3.71034496391760758659681411813, −3.26772768895654717494045835404, −3.20590131121305948714281592966, −3.13307023314842390052845085469, −2.35681549853866938015419789656, −2.18962633727294920515719389228, −1.88793306061726811368663177046, −1.86039105631635865065620628667, −1.38160236323605464931800724487, −0.880392108668189233234294255987, −0.080205774171880955124001131914,
0.080205774171880955124001131914, 0.880392108668189233234294255987, 1.38160236323605464931800724487, 1.86039105631635865065620628667, 1.88793306061726811368663177046, 2.18962633727294920515719389228, 2.35681549853866938015419789656, 3.13307023314842390052845085469, 3.20590131121305948714281592966, 3.26772768895654717494045835404, 3.71034496391760758659681411813, 4.14747976697602489594439922433, 4.18133568967525758006925552893, 4.33349365851407419642772930446, 4.84860549579251025626369382622, 4.87359054721853402831687773341, 5.09340614129438693617138393357, 5.22240350698829095250257634884, 5.67682839751901155424779844417, 5.88215485265376661954309589454, 6.00630590992215105568389779806, 6.23408767164125756187837618587, 6.76612360278980203045233818831, 6.80965300056167292998699194698, 6.88318775903522271829812548419
