| L(s) = 1 | − 497·4-s − 7.09e4·11-s + 8.77e4·16-s + 8.06e5·19-s − 1.49e5·29-s − 1.00e7·31-s − 2.84e7·41-s + 3.52e7·44-s + 8.35e7·49-s + 3.75e8·59-s + 3.08e8·61-s − 9.47e7·64-s + 4.56e8·71-s − 4.00e8·76-s + 1.86e9·79-s + 4.49e8·89-s − 6.98e8·101-s + 5.89e9·109-s + 7.41e7·116-s + 2.72e9·121-s + 4.99e9·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 0.970·4-s − 1.46·11-s + 0.334·16-s + 1.41·19-s − 0.0391·29-s − 1.95·31-s − 1.57·41-s + 1.41·44-s + 2.06·49-s + 4.03·59-s + 2.84·61-s − 0.705·64-s + 2.13·71-s − 1.37·76-s + 5.38·79-s + 0.758·89-s − 0.667·101-s + 3.99·109-s + 0.0380·116-s + 1.15·121-s + 1.89·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(5.300246005\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.300246005\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 + 497 T^{2} + 9953 p^{4} T^{4} + 497 p^{18} T^{6} + p^{36} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 1704636 p^{2} T^{2} + 38066346502 p^{6} T^{4} - 1704636 p^{20} T^{6} + p^{36} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 35488 T + 526013014 T^{2} + 35488 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 27567505292 T^{2} + \)\(36\!\cdots\!18\)\( T^{4} - 27567505292 p^{18} T^{6} + p^{36} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 388734753820 T^{2} + \)\(65\!\cdots\!18\)\( T^{4} - 388734753820 p^{18} T^{6} + p^{36} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 403296 T + 684929514838 T^{2} - 403296 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 2800589695204 T^{2} + \)\(81\!\cdots\!58\)\( T^{4} + 2800589695204 p^{18} T^{6} + p^{36} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 74572 T + 28833430018078 T^{2} + 74572 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 5027128 T + 52415931233342 T^{2} + 5027128 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 433247572021484 T^{2} + \)\(79\!\cdots\!78\)\( T^{4} - 433247572021484 p^{18} T^{6} + p^{36} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 14211332 T + 443988635955862 T^{2} + 14211332 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 1570463388232460 T^{2} + \)\(11\!\cdots\!98\)\( T^{4} - 1570463388232460 p^{18} T^{6} + p^{36} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 568239172667780 T^{2} + \)\(55\!\cdots\!78\)\( T^{4} + 568239172667780 p^{18} T^{6} + p^{36} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 9086956380170956 T^{2} + \)\(38\!\cdots\!18\)\( T^{4} - 9086956380170956 p^{18} T^{6} + p^{36} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 187863136 T + 23071633420288438 T^{2} - 187863136 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 154080060 T + 23302683905802238 T^{2} - 154080060 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 22768078838174100 T^{2} + \)\(15\!\cdots\!18\)\( T^{4} + 22768078838174100 p^{18} T^{6} + p^{36} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 228270976 T + 45777616900481806 T^{2} - 228270976 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 135645128086811996 T^{2} + \)\(11\!\cdots\!58\)\( T^{4} - 135645128086811996 p^{18} T^{6} + p^{36} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 932406760 T + 453226630902929438 T^{2} - 932406760 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 702700996120787372 T^{2} + \)\(19\!\cdots\!38\)\( T^{4} - 702700996120787372 p^{18} T^{6} + p^{36} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 2522676 p T + 610925899926766678 T^{2} - 2522676 p^{10} T^{3} + p^{18} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 1619811253917303940 T^{2} + \)\(14\!\cdots\!78\)\( T^{4} + 1619811253917303940 p^{18} T^{6} + p^{36} T^{8} \) |
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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
−7.36697470864644085938755213570, −7.05614443826670125199238894821, −6.87219043576745223185795965241, −6.37475281287946740625617427575, −6.30519103577572713151334370894, −5.71321097447361480522226150588, −5.51082946275169540595181698376, −5.24298258033867187651716844112, −5.12820089723614580283508316070, −5.08826264288552096409262559836, −4.65642657817789885464537507733, −4.13007854635118792960722248376, −3.93444548541564001248007638272, −3.47364006490433815397166686034, −3.46698698930746329628580526510, −3.39042209510176586209561939356, −2.60289532414162816697624095305, −2.26021993834447428482552262177, −2.14205520848433895364177083933, −2.11383970481112690539673228553, −1.37214266540901564084775857721, −0.971971180180974662979920835451, −0.63493116314670924548909460727, −0.56924527745812172637927909249, −0.34523772384080472534054126602,
0.34523772384080472534054126602, 0.56924527745812172637927909249, 0.63493116314670924548909460727, 0.971971180180974662979920835451, 1.37214266540901564084775857721, 2.11383970481112690539673228553, 2.14205520848433895364177083933, 2.26021993834447428482552262177, 2.60289532414162816697624095305, 3.39042209510176586209561939356, 3.46698698930746329628580526510, 3.47364006490433815397166686034, 3.93444548541564001248007638272, 4.13007854635118792960722248376, 4.65642657817789885464537507733, 5.08826264288552096409262559836, 5.12820089723614580283508316070, 5.24298258033867187651716844112, 5.51082946275169540595181698376, 5.71321097447361480522226150588, 6.30519103577572713151334370894, 6.37475281287946740625617427575, 6.87219043576745223185795965241, 7.05614443826670125199238894821, 7.36697470864644085938755213570
