| L(s) = 1 | + 4·4-s + 12·16-s − 48·17-s − 32·19-s − 60·23-s + 32·31-s − 60·47-s + 128·49-s + 96·53-s − 76·61-s + 32·64-s − 192·68-s − 128·76-s + 160·79-s − 168·83-s − 240·92-s + 204·107-s − 416·109-s + 192·113-s + 466·121-s + 128·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 4-s + 3/4·16-s − 2.82·17-s − 1.68·19-s − 2.60·23-s + 1.03·31-s − 1.27·47-s + 2.61·49-s + 1.81·53-s − 1.24·61-s + 1/2·64-s − 2.82·68-s − 1.68·76-s + 2.02·79-s − 2.02·83-s − 2.60·92-s + 1.90·107-s − 3.81·109-s + 1.69·113-s + 3.85·121-s + 1.03·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.082945276\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.082945276\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 128 T^{2} + 7746 T^{4} - 128 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 233 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 482 T^{2} + 108003 T^{4} - 482 p^{4} T^{6} + p^{8} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 24 T + 704 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 16 T + 498 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 + 30 T + 1211 T^{2} + 30 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 1636 T^{2} + 1420134 T^{4} - 1636 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 16 T + 1536 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 3938 T^{2} + 7311651 T^{4} - 3938 p^{4} T^{6} + p^{8} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 3232 T^{2} + 5930178 T^{4} - 3232 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 5252 T^{2} + 12830310 T^{4} - 5252 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 30 T + 4571 T^{2} + 30 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 - 48 T + 3602 T^{2} - 48 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 6418 T^{2} + 22546995 T^{4} - 6418 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 38 T + 1971 T^{2} + 38 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 1952 T^{2} + 39222690 T^{4} - 1952 p^{4} T^{6} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 6898 T^{2} + 60829395 T^{4} - 6898 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 9860 T^{2} + 48587334 T^{4} - 9860 p^{4} T^{6} + p^{8} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 80 T + 12624 T^{2} - 80 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 + 84 T + 10934 T^{2} + 84 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 29056 T^{2} + 336453954 T^{4} - 29056 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 478 T^{2} + 131035683 T^{4} + 478 p^{4} T^{6} + p^{8} T^{8} \) |
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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
−6.54469325864132113098721549413, −6.35122681686514805360759835364, −6.31221228378096393176288159404, −6.23160821531253209253982653661, −5.73853410997878330776551184610, −5.68324079201999124297651976887, −5.42544003059557418337337580112, −5.24670354929068785893960962510, −4.65620640424977306473020931995, −4.55593459641817771530558495523, −4.32079030283158571434223166656, −4.17395062331412432166886001496, −3.97344483906806115703096198063, −3.96826654382944837906687981682, −3.17763442481011472753675446494, −3.03733298256672182018787042112, −3.00215452435156416723292973004, −2.46385684939544420821773764802, −2.10519469394703685136929402883, −1.97375067964315425316860291678, −1.94368917537959599831056623092, −1.75722475395894680401400536706, −0.836530008868899030653961388961, −0.62443306090531714170607353627, −0.27748471026910730764083042583,
0.27748471026910730764083042583, 0.62443306090531714170607353627, 0.836530008868899030653961388961, 1.75722475395894680401400536706, 1.94368917537959599831056623092, 1.97375067964315425316860291678, 2.10519469394703685136929402883, 2.46385684939544420821773764802, 3.00215452435156416723292973004, 3.03733298256672182018787042112, 3.17763442481011472753675446494, 3.96826654382944837906687981682, 3.97344483906806115703096198063, 4.17395062331412432166886001496, 4.32079030283158571434223166656, 4.55593459641817771530558495523, 4.65620640424977306473020931995, 5.24670354929068785893960962510, 5.42544003059557418337337580112, 5.68324079201999124297651976887, 5.73853410997878330776551184610, 6.23160821531253209253982653661, 6.31221228378096393176288159404, 6.35122681686514805360759835364, 6.54469325864132113098721549413
