| L(s) = 1 | + 432·7-s + 4.35e3·13-s − 2.69e4·19-s − 2.35e4·25-s + 5.61e4·31-s + 2.37e5·37-s + 2.84e3·43-s − 1.86e5·49-s − 1.46e5·61-s + 1.54e6·67-s + 2.81e5·73-s − 1.25e6·79-s + 1.88e6·91-s − 1.32e6·97-s + 3.92e6·103-s + 3.04e6·109-s + 2.98e6·121-s + 127-s + 131-s − 1.16e7·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 1.25·7-s + 1.98·13-s − 3.92·19-s − 1.50·25-s + 1.88·31-s + 4.67·37-s + 0.0358·43-s − 1.58·49-s − 0.646·61-s + 5.14·67-s + 0.724·73-s − 2.55·79-s + 2.49·91-s − 1.44·97-s + 3.59·103-s + 2.35·109-s + 1.68·121-s − 4.94·133-s + 0.148·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+3)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.377434855\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.377434855\) |
| \(L(4)\) |
|
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 | $D_4\times C_2$ | \( 1 + 4704 p T^{2} + 16711154 p^{2} T^{4} + 4704 p^{13} T^{6} + p^{24} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 - 216 T + 163442 T^{2} - 216 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 271020 p T^{2} + 5196482322662 T^{4} - 271020 p^{13} T^{6} + p^{24} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 2176 T + 6744882 T^{2} - 2176 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 5032320 p T^{2} + 2991357595693442 T^{4} - 5032320 p^{13} T^{6} + p^{24} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 13472 T + 134120178 T^{2} + 13472 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 45666180 T^{2} - 26156437245811258 T^{4} - 45666180 p^{12} T^{6} + p^{24} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 1763439520 T^{2} + 1394728586290536162 T^{4} - 1763439520 p^{12} T^{6} + p^{24} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 28088 T + 538954578 T^{2} - 28088 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 118500 T + 8159603798 T^{2} - 118500 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 8906980864 T^{2} + 39521221186647918786 T^{4} - 8906980864 p^{12} T^{6} + p^{24} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 1424 T - 9422416878 T^{2} - 1424 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 25582810180 T^{2} + \)\(32\!\cdots\!02\)\( T^{4} - 25582810180 p^{12} T^{6} + p^{24} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 20589242016 T^{2} + \)\(92\!\cdots\!46\)\( T^{4} - 20589242016 p^{12} T^{6} + p^{24} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 122630525220 T^{2} + \)\(71\!\cdots\!42\)\( T^{4} - 122630525220 p^{12} T^{6} + p^{24} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 73412 T + 78569040438 T^{2} + 73412 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 773104 T + 322721854962 T^{2} - 773104 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 331438972164 T^{2} + \)\(56\!\cdots\!06\)\( T^{4} - 331438972164 p^{12} T^{6} + p^{24} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 140960 T + 281068505058 T^{2} - 140960 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 629240 T + 490074054162 T^{2} + 629240 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 1204278186660 T^{2} + \)\(57\!\cdots\!02\)\( T^{4} - 1204278186660 p^{12} T^{6} + p^{24} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 1438573805824 T^{2} + \)\(93\!\cdots\!86\)\( T^{4} - 1438573805824 p^{12} T^{6} + p^{24} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 661184 T + 1313995880322 T^{2} + 661184 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
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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
−6.44330637946034709346151269268, −6.43746648122883986841291826944, −6.27495070043221710129375418132, −5.97753071864223334262342425765, −5.97559009236825110042280300400, −5.83987133508109933685718627314, −5.15638396120402968050215749136, −4.82307342601328866061825817306, −4.67604405220804018418020977977, −4.65641827102308335780059634301, −4.28437316719177645285057565176, −3.98786392041609553118494878705, −3.79536783888621239138183250773, −3.64001343365333164285156897350, −3.31861526800040200989955098566, −2.62786203169559388486922928483, −2.51684223247330565995701995015, −2.35680753518360358866045929664, −1.96099852836927451472852615224, −1.94445966331318479349891136725, −1.19544757760235940545307121448, −1.09958993919252629780147871204, −1.08163723846001968626134288978, −0.48180019635429231915547943838, −0.15214415930136996305545132979,
0.15214415930136996305545132979, 0.48180019635429231915547943838, 1.08163723846001968626134288978, 1.09958993919252629780147871204, 1.19544757760235940545307121448, 1.94445966331318479349891136725, 1.96099852836927451472852615224, 2.35680753518360358866045929664, 2.51684223247330565995701995015, 2.62786203169559388486922928483, 3.31861526800040200989955098566, 3.64001343365333164285156897350, 3.79536783888621239138183250773, 3.98786392041609553118494878705, 4.28437316719177645285057565176, 4.65641827102308335780059634301, 4.67604405220804018418020977977, 4.82307342601328866061825817306, 5.15638396120402968050215749136, 5.83987133508109933685718627314, 5.97559009236825110042280300400, 5.97753071864223334262342425765, 6.27495070043221710129375418132, 6.43746648122883986841291826944, 6.44330637946034709346151269268
