| L(s) = 1 | − 8·7-s + 32·13-s − 32·19-s + 72·25-s + 120·31-s + 88·37-s − 144·43-s + 36·49-s + 200·61-s − 112·67-s + 272·73-s + 488·79-s − 256·91-s + 160·97-s + 40·103-s + 192·109-s + 372·121-s + 127-s + 131-s + 256·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 8/7·7-s + 2.46·13-s − 1.68·19-s + 2.87·25-s + 3.87·31-s + 2.37·37-s − 3.34·43-s + 0.734·49-s + 3.27·61-s − 1.67·67-s + 3.72·73-s + 6.17·79-s − 2.81·91-s + 1.64·97-s + 0.388·103-s + 1.76·109-s + 3.07·121-s + 0.00787·127-s + 0.00763·131-s + 1.92·133-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(9.914162348\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.914162348\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 72 T^{2} + 98 p^{2} T^{4} - 72 p^{4} T^{6} + p^{8} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 + 4 T + 6 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 372 T^{2} + 62342 T^{4} - 372 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 16 T + 186 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 288 T^{2} + 184322 T^{4} - 288 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 16 T + 690 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 858 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 2152 T^{2} + 2530002 T^{4} - 2152 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 60 T + 2726 T^{2} - 60 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 44 T + 2838 T^{2} - 44 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 2560 T^{2} + 3056322 T^{4} - 2560 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 72 T + 3458 T^{2} + 72 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 6900 T^{2} + 21047462 T^{4} - 6900 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 7720 T^{2} + 28930962 T^{4} - 7720 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 1500 T^{2} + 2678822 T^{4} + 1500 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 100 T + 8406 T^{2} - 100 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 56 T + 9666 T^{2} + 56 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 5044 T^{2} + 16873062 T^{4} - 5044 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 136 T + 14898 T^{2} - 136 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 244 T + 26502 T^{2} - 244 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 1484 T^{2} - 19009338 T^{4} + 1484 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 11712 T^{2} + 138025922 T^{4} - 11712 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 80 T + 4194 T^{2} - 80 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−6.60254571919395091490225932391, −6.44756765501538743747378083602, −6.34316156392988476286734484842, −6.29693974714068648246736462153, −6.21962193758682821396390053536, −5.89730636616002723261691303036, −5.29525766982348591059551325014, −5.17495139039200293657770062535, −5.07427648830470285342561506383, −4.70509501026542093917886100050, −4.50930034401196248040163480042, −4.23107456992774786149850739134, −4.11851923872372311323664836202, −3.56235614940779048055702393723, −3.49606165359999204617861739657, −3.20539105197509127788600172019, −3.12383612065350244356248261814, −2.80707541237407016429972654944, −2.18724792164058654673041289687, −2.14715592456740725073104647783, −2.00529509394838221187036970604, −1.16101150000073812417668539232, −0.813123870968506483201611632886, −0.72743883995929549268976671204, −0.65950472578928345040594991349,
0.65950472578928345040594991349, 0.72743883995929549268976671204, 0.813123870968506483201611632886, 1.16101150000073812417668539232, 2.00529509394838221187036970604, 2.14715592456740725073104647783, 2.18724792164058654673041289687, 2.80707541237407016429972654944, 3.12383612065350244356248261814, 3.20539105197509127788600172019, 3.49606165359999204617861739657, 3.56235614940779048055702393723, 4.11851923872372311323664836202, 4.23107456992774786149850739134, 4.50930034401196248040163480042, 4.70509501026542093917886100050, 5.07427648830470285342561506383, 5.17495139039200293657770062535, 5.29525766982348591059551325014, 5.89730636616002723261691303036, 6.21962193758682821396390053536, 6.29693974714068648246736462153, 6.34316156392988476286734484842, 6.44756765501538743747378083602, 6.60254571919395091490225932391
