| L(s) = 1 | + 728·5-s − 1.26e4·13-s + 3.91e5·17-s − 9.11e5·25-s + 7.05e5·29-s + 4.44e6·37-s − 2.95e6·41-s + 1.29e7·49-s + 4.50e6·53-s − 4.01e7·61-s − 9.19e6·65-s − 5.92e6·73-s + 2.85e8·85-s − 1.32e8·89-s + 5.64e7·97-s + 2.31e8·101-s + 5.57e8·109-s + 4.99e8·113-s + 6.21e8·121-s − 9.52e8·125-s + ⋯ |
| L(s) = 1 | + 1.16·5-s − 0.442·13-s + 4.69·17-s − 2.33·25-s + 0.997·29-s + 2.37·37-s − 1.04·41-s + 2.24·49-s + 0.570·53-s − 2.90·61-s − 0.515·65-s − 0.208·73-s + 5.46·85-s − 2.11·89-s + 0.637·97-s + 2.22·101-s + 3.94·109-s + 3.06·113-s + 2.90·121-s − 3.90·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+4)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(14.58730974\) |
| \(L(\frac12)\) |
\(\approx\) |
\(14.58730974\) |
| \(L(5)\) |
|
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 | $D_{4}$ | \( ( 1 - 364 T + 130926 p T^{2} - 364 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 12949124 T^{2} + 2124073815270 p^{2} T^{4} - 12949124 p^{16} T^{6} + p^{32} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 621783908 T^{2} + 181770025950389574 T^{4} - 621783908 p^{16} T^{6} + p^{32} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 6316 T - 471179994 T^{2} + 6316 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 - 195996 T + 21760239302 T^{2} - 195996 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 10037112676 T^{2} + \)\(45\!\cdots\!06\)\( T^{4} - 10037112676 p^{16} T^{6} + p^{32} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 230553122948 T^{2} + \)\(25\!\cdots\!54\)\( T^{4} - 230553122948 p^{16} T^{6} + p^{32} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 352748 T + 999396063654 T^{2} - 352748 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 1561001128964 T^{2} + \)\(12\!\cdots\!10\)\( p^{2} T^{4} - 1561001128964 p^{16} T^{6} + p^{32} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 2221524 T + 8217364493222 T^{2} - 2221524 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 1476676 T + 2025162183942 T^{2} + 1476676 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 9479410588004 T^{2} + \)\(25\!\cdots\!90\)\( T^{4} - 9479410588004 p^{16} T^{6} + p^{32} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 32334216312836 T^{2} + \)\(95\!\cdots\!66\)\( T^{4} - 32334216312836 p^{16} T^{6} + p^{32} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 2250924 T + 123936144680870 T^{2} - 2250924 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 305254750575716 T^{2} + \)\(59\!\cdots\!46\)\( T^{4} - 305254750575716 p^{16} T^{6} + p^{32} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 20079532 T + 283245594359334 T^{2} + 20079532 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 1540404309460580 T^{2} + \)\(92\!\cdots\!98\)\( T^{4} - 1540404309460580 p^{16} T^{6} + p^{32} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 783735682299524 T^{2} + \)\(23\!\cdots\!50\)\( T^{4} - 783735682299524 p^{16} T^{6} + p^{32} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 2960412 T + 1505171737656134 T^{2} + 2960412 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 4755170211804164 T^{2} + \)\(98\!\cdots\!70\)\( T^{4} - 4755170211804164 p^{16} T^{6} + p^{32} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 5209133236219748 T^{2} + \)\(14\!\cdots\!74\)\( T^{4} - 5209133236219748 p^{16} T^{6} + p^{32} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 66323940 T + 6939976370286662 T^{2} + 66323940 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 28222436 T + 13755128655195846 T^{2} - 28222436 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
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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
−7.28376067988661618941714309153, −7.10948207956894547551520112571, −6.32649981595669051362449166700, −6.25077015172947334576103895374, −6.07724991195879795084092903031, −5.74956779645732532705955288065, −5.66805944187657280215863405325, −5.36751609125962511437993008076, −5.36676015915262584933321223302, −4.64660382932874412077010562309, −4.54054769389466582498012500415, −4.20980981612918985346027768168, −3.99228496240474911606966672119, −3.35812249711331330950907291933, −3.20613480544950991256855186236, −3.11961397685040269911936223127, −2.96313896395331317369780066053, −2.13362166571547993771128887386, −2.09614382620415974845065764374, −1.90766330975235832958936443624, −1.49448128609019521772906587136, −1.04840096545093274352028884994, −0.845843258059997347110128009766, −0.69459831318461895127260168633, −0.33844341703226508853590705427,
0.33844341703226508853590705427, 0.69459831318461895127260168633, 0.845843258059997347110128009766, 1.04840096545093274352028884994, 1.49448128609019521772906587136, 1.90766330975235832958936443624, 2.09614382620415974845065764374, 2.13362166571547993771128887386, 2.96313896395331317369780066053, 3.11961397685040269911936223127, 3.20613480544950991256855186236, 3.35812249711331330950907291933, 3.99228496240474911606966672119, 4.20980981612918985346027768168, 4.54054769389466582498012500415, 4.64660382932874412077010562309, 5.36676015915262584933321223302, 5.36751609125962511437993008076, 5.66805944187657280215863405325, 5.74956779645732532705955288065, 6.07724991195879795084092903031, 6.25077015172947334576103895374, 6.32649981595669051362449166700, 7.10948207956894547551520112571, 7.28376067988661618941714309153
