| L(s) = 1 | + 32·13-s − 80·19-s − 16·25-s + 16·31-s − 152·37-s − 80·43-s + 14·49-s − 232·61-s + 192·67-s − 96·73-s + 304·79-s − 288·97-s + 272·103-s + 288·109-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 188·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 2.46·13-s − 4.21·19-s − 0.639·25-s + 0.516·31-s − 4.10·37-s − 1.86·43-s + 2/7·49-s − 3.80·61-s + 2.86·67-s − 1.31·73-s + 3.84·79-s − 2.96·97-s + 2.64·103-s + 2.64·109-s + 0.165·121-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 + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 1.11·169-s + 0.00578·173-s + 0.00558·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.548991871\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.548991871\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 16 T^{2} + 866 T^{4} + 16 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 22214 T^{4} - 20 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 16 T + 290 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 368 T^{2} + 125186 T^{4} - 368 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $C_2$ | \( ( 1 + 20 T + p^{2} T^{2} )^{4} \) |
| 23 | $D_4\times C_2$ | \( 1 - 1652 T^{2} + 1234790 T^{4} - 1652 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 1472 T^{2} + 1309346 T^{4} - 1472 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 8 T + 1490 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 38 T + p^{2} T^{2} )^{4} \) |
| 41 | $D_4\times C_2$ | \( 1 - 2800 T^{2} + 4672194 T^{4} - 2800 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 40 T + 66 T^{2} + 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 4130 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 + 4928 T^{2} + 21271650 T^{4} + 4928 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 10532 T^{2} + 49098278 T^{4} - 10532 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 116 T + 9014 T^{2} + 116 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 96 T + 4114 T^{2} - 96 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 19700 T^{2} + 147838694 T^{4} - 19700 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 48 T + 8434 T^{2} + 48 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 152 T + 16466 T^{2} - 152 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 9380 T^{2} + 87552614 T^{4} - 9380 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 12208 T^{2} + 68358210 T^{4} - 12208 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 144 T + 10450 T^{2} + 144 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−6.01835964659609836451912685583, −5.52201177982273899771831873180, −5.42458489126571379834774246313, −5.20385039673319387007013983499, −4.90941232164210381822519433620, −4.81883338750728410551201701511, −4.65113826151124622572342536638, −4.18587943126388283080875211853, −4.13735321757102972119277301514, −4.12183189517969045943921005133, −3.73460274532662397689047482937, −3.53452980492984024759592298427, −3.36067065923157977044311261685, −3.13679968233974593969398506987, −3.12503203666539465076982390309, −2.59622242783980235021720706738, −2.23964727222917821997494584481, −1.94480625124304014274870119777, −1.85837865663615916081071298396, −1.84038682333968100088694703488, −1.61200068733452187481589994226, −0.977512718964534596528642118833, −0.921733791934033855527297296176, −0.29335694448670033507844385990, −0.22369898245173927290370299681,
0.22369898245173927290370299681, 0.29335694448670033507844385990, 0.921733791934033855527297296176, 0.977512718964534596528642118833, 1.61200068733452187481589994226, 1.84038682333968100088694703488, 1.85837865663615916081071298396, 1.94480625124304014274870119777, 2.23964727222917821997494584481, 2.59622242783980235021720706738, 3.12503203666539465076982390309, 3.13679968233974593969398506987, 3.36067065923157977044311261685, 3.53452980492984024759592298427, 3.73460274532662397689047482937, 4.12183189517969045943921005133, 4.13735321757102972119277301514, 4.18587943126388283080875211853, 4.65113826151124622572342536638, 4.81883338750728410551201701511, 4.90941232164210381822519433620, 5.20385039673319387007013983499, 5.42458489126571379834774246313, 5.52201177982273899771831873180, 6.01835964659609836451912685583
