| L(s) = 1 | + 6.65e3·9-s − 8.68e3·11-s + 3.64e4·19-s − 1.11e5·29-s + 6.03e5·31-s − 2.16e5·41-s + 8.24e5·49-s + 4.13e6·59-s + 1.16e6·61-s + 9.45e6·71-s − 1.43e7·79-s + 2.46e7·81-s + 1.19e7·89-s − 5.77e7·99-s − 5.96e6·101-s − 6.96e5·109-s − 1.04e7·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.06e7·169-s + ⋯ |
| L(s) = 1 | + 3.04·9-s − 1.96·11-s + 1.21·19-s − 0.849·29-s + 3.63·31-s − 0.491·41-s + 1.00·49-s + 2.62·59-s + 0.656·61-s + 3.13·71-s − 3.27·79-s + 5.15·81-s + 1.80·89-s − 5.98·99-s − 0.576·101-s − 0.0514·109-s − 0.538·121-s + 0.648·169-s + 3.70·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.736966242\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.736966242\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 6650 T^{2} + 2175907 p^{2} T^{4} - 6650 p^{14} T^{6} + p^{28} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 824500 T^{2} + 1114255248198 T^{4} - 824500 p^{14} T^{6} + p^{28} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 4344 T + 33551301 T^{2} + 4344 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 40685300 T^{2} - 153894343380522 T^{4} - 40685300 p^{14} T^{6} + p^{28} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 1456928850 T^{2} + 863620794440640083 T^{4} - 1456928850 p^{14} T^{6} + p^{28} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 18200 T + 1670645253 T^{2} - 18200 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 1310008300 T^{2} + 20334801990676168518 T^{4} + 1310008300 p^{14} T^{6} + p^{28} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 55800 T + 7178799018 T^{2} + 55800 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 301776 T + 74506854266 T^{2} - 301776 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 152649548300 T^{2} + \)\(16\!\cdots\!78\)\( T^{4} - 152649548300 p^{14} T^{6} + p^{28} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 2646 p T + 391801850811 T^{2} + 2646 p^{8} T^{3} + p^{14} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 489365211500 T^{2} + \)\(14\!\cdots\!98\)\( T^{4} - 489365211500 p^{14} T^{6} + p^{28} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 367649295900 T^{2} + \)\(45\!\cdots\!38\)\( T^{4} - 367649295900 p^{14} T^{6} + p^{28} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 4060694649900 T^{2} + \)\(68\!\cdots\!38\)\( T^{4} - 4060694649900 p^{14} T^{6} + p^{28} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 2067600 T + 4803250907238 T^{2} - 2067600 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 582044 T - 1843726773474 T^{2} - 582044 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 22459694871850 T^{2} + \)\(19\!\cdots\!83\)\( T^{4} - 22459694871850 p^{14} T^{6} + p^{28} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 4728216 T + 23373621952446 T^{2} - 4728216 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 40979274242450 T^{2} + \)\(66\!\cdots\!43\)\( T^{4} - 40979274242450 p^{14} T^{6} + p^{28} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 7186200 T + 50054286054218 T^{2} + 7186200 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 34386337777850 T^{2} + \)\(16\!\cdots\!83\)\( T^{4} - 34386337777850 p^{14} T^{6} + p^{28} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 5990850 T + 65725956363283 T^{2} - 5990850 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 171757286723900 T^{2} + \)\(19\!\cdots\!38\)\( T^{4} - 171757286723900 p^{14} T^{6} + p^{28} T^{8} \) |
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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
−6.92147875817446264716624949196, −6.89839613268150017995275999087, −6.59925584991982836269282092443, −6.31639226342247909446124480724, −5.79903955587755033231814099990, −5.70132613845018747729340362789, −5.46401823791531526764301846424, −4.87961354147067696825609167587, −4.87785857881321348783656769617, −4.69472066207792124140753021454, −4.68979550525214890788436496334, −3.86571575898702183249577208370, −3.85715091047571717406023538555, −3.62434812833550369193651627020, −3.55217984611823513131736860767, −2.66087952485680995455964920106, −2.45882792298180995653344575380, −2.44733903498248118252777141121, −2.41516796544104538328712483464, −1.58657825388125791773333991261, −1.23505076424872810836719332979, −1.16069707680663928644258197141, −1.06458617702928367072439025346, −0.57212415822337078808622728890, −0.11675091725878755086338452559,
0.11675091725878755086338452559, 0.57212415822337078808622728890, 1.06458617702928367072439025346, 1.16069707680663928644258197141, 1.23505076424872810836719332979, 1.58657825388125791773333991261, 2.41516796544104538328712483464, 2.44733903498248118252777141121, 2.45882792298180995653344575380, 2.66087952485680995455964920106, 3.55217984611823513131736860767, 3.62434812833550369193651627020, 3.85715091047571717406023538555, 3.86571575898702183249577208370, 4.68979550525214890788436496334, 4.69472066207792124140753021454, 4.87785857881321348783656769617, 4.87961354147067696825609167587, 5.46401823791531526764301846424, 5.70132613845018747729340362789, 5.79903955587755033231814099990, 6.31639226342247909446124480724, 6.59925584991982836269282092443, 6.89839613268150017995275999087, 6.92147875817446264716624949196
