| L(s) = 1 | − 16·7-s − 8·13-s − 32·19-s − 8·31-s + 136·37-s + 80·43-s + 144·49-s − 40·61-s − 304·67-s − 152·73-s − 200·79-s + 128·91-s + 424·97-s − 112·103-s + 104·109-s + 88·121-s + 127-s + 131-s + 512·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 456·169-s + ⋯ |
| L(s) = 1 | − 2.28·7-s − 0.615·13-s − 1.68·19-s − 0.258·31-s + 3.67·37-s + 1.86·43-s + 2.93·49-s − 0.655·61-s − 4.53·67-s − 2.08·73-s − 2.53·79-s + 1.40·91-s + 4.37·97-s − 1.08·103-s + 0.954·109-s + 8/11·121-s + 0.00787·127-s + 0.00763·131-s + 3.84·133-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 − 2.69·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1664603776\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1664603776\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $D_{4}$ | \( ( 1 + 8 T + 24 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 8 p T^{2} + 18258 T^{4} - 8 p^{5} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 4 T + 252 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 290 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 16 T + 426 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 1612 T^{2} + 1157478 T^{4} - 1612 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 700 T^{2} + 707622 T^{4} - 700 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 4 T + 1566 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 68 T + 3084 T^{2} - 68 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 640 T^{2} + 3887682 T^{4} - 640 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 40 T + 3738 T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 340 T^{2} - 372378 T^{4} - 340 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 1876 T^{2} - 4075194 T^{4} - 1876 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 280 T^{2} - 9454638 T^{4} - 280 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 20 T + 4302 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 76 T + p^{2} T^{2} )^{4} \) |
| 71 | $D_4\times C_2$ | \( 1 - 13540 T^{2} + 89191302 T^{4} - 13540 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 76 T + 8862 T^{2} + 76 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 100 T + 11742 T^{2} + 100 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 21652 T^{2} + 211289478 T^{4} - 21652 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 14368 T^{2} + 154232898 T^{4} - 14368 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 212 T + 29694 T^{2} - 212 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−5.89981147808746665647165110752, −5.88074199798297924241705593697, −5.73828375630187579236408718718, −5.43182518324316677362814973518, −4.84260700436234128976182806429, −4.67883655695569554359389073053, −4.67730264170461972576771444745, −4.50747630967938146846093644902, −4.26367909135738346686302967019, −4.06965983306130057801523154073, −3.83006857001908826573049304447, −3.48988079727488196726960259151, −3.46587591358928062635254786761, −2.97632856003991694980726257195, −2.96337392667676076825959877931, −2.76842498517207105295229672494, −2.44475434725568436289809054309, −2.26591490869014871557559202262, −2.20905949166861389620545552324, −1.69027522279004264595772195596, −1.24989833817534671668105779135, −1.05675586312280959665979588893, −0.923912186432639386413852387468, −0.20431381529967858563518611151, −0.11787401743627227566642307129,
0.11787401743627227566642307129, 0.20431381529967858563518611151, 0.923912186432639386413852387468, 1.05675586312280959665979588893, 1.24989833817534671668105779135, 1.69027522279004264595772195596, 2.20905949166861389620545552324, 2.26591490869014871557559202262, 2.44475434725568436289809054309, 2.76842498517207105295229672494, 2.96337392667676076825959877931, 2.97632856003991694980726257195, 3.46587591358928062635254786761, 3.48988079727488196726960259151, 3.83006857001908826573049304447, 4.06965983306130057801523154073, 4.26367909135738346686302967019, 4.50747630967938146846093644902, 4.67730264170461972576771444745, 4.67883655695569554359389073053, 4.84260700436234128976182806429, 5.43182518324316677362814973518, 5.73828375630187579236408718718, 5.88074199798297924241705593697, 5.89981147808746665647165110752
