| L(s) = 1 | + 75·4-s − 8.68e3·11-s + 7.95e3·16-s − 3.64e4·19-s + 1.11e5·29-s − 6.03e5·31-s + 2.16e5·41-s − 6.51e5·44-s + 8.24e5·49-s + 4.13e6·59-s + 1.16e6·61-s + 2.00e6·64-s + 9.45e6·71-s − 2.73e6·76-s + 1.43e7·79-s − 1.19e7·89-s + 5.96e6·101-s − 6.96e5·109-s + 8.37e6·116-s − 1.04e7·121-s − 4.52e7·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 0.585·4-s − 1.96·11-s + 0.485·16-s − 1.21·19-s + 0.849·29-s − 3.63·31-s + 0.491·41-s − 1.15·44-s + 1.00·49-s + 2.62·59-s + 0.656·61-s + 0.953·64-s + 3.13·71-s − 0.713·76-s + 3.27·79-s − 1.80·89-s + 0.576·101-s − 0.0514·109-s + 0.497·116-s − 0.538·121-s − 2.13·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.8334806391\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8334806391\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 - 75 T^{2} - 583 p^{2} T^{4} - 75 p^{14} T^{6} + p^{28} T^{8} \) |
| 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
−7.87835257401863553233825610769, −7.09241296675523203945745427393, −7.04625670768647129752550417809, −6.89359517901211463157980489763, −6.87910717191725929972502354007, −6.10167405734846009719046181572, −6.03745570170226983135167550311, −5.69421882024093727240998652775, −5.28806290215749556256824839031, −5.25154131430515927126042961176, −5.11714971844296285389790663667, −4.58218610920647262199319683073, −4.11340680889502330877220752065, −3.98864921472166742896197310927, −3.49153608864423811503474324158, −3.44728791659691336190256581970, −2.99692825532757400952418766329, −2.36466366597917702536087308102, −2.26464758525012299215418207204, −2.18754100571432142693586050688, −1.92883733827878584522542880719, −1.13049070306634862382129361764, −0.965746962726383026959837261708, −0.51649703080345255155017176975, −0.12230453031373761857531107956,
0.12230453031373761857531107956, 0.51649703080345255155017176975, 0.965746962726383026959837261708, 1.13049070306634862382129361764, 1.92883733827878584522542880719, 2.18754100571432142693586050688, 2.26464758525012299215418207204, 2.36466366597917702536087308102, 2.99692825532757400952418766329, 3.44728791659691336190256581970, 3.49153608864423811503474324158, 3.98864921472166742896197310927, 4.11340680889502330877220752065, 4.58218610920647262199319683073, 5.11714971844296285389790663667, 5.25154131430515927126042961176, 5.28806290215749556256824839031, 5.69421882024093727240998652775, 6.03745570170226983135167550311, 6.10167405734846009719046181572, 6.87910717191725929972502354007, 6.89359517901211463157980489763, 7.04625670768647129752550417809, 7.09241296675523203945745427393, 7.87835257401863553233825610769
