| L(s) = 1 | + 48·4-s + 24·11-s + 796·16-s − 8.42e3·19-s + 8.13e3·29-s − 5.19e3·31-s − 2.24e4·41-s + 1.15e3·44-s + 2.63e4·49-s − 1.27e5·59-s + 1.46e4·61-s + 1.49e4·64-s − 1.96e5·71-s − 4.04e5·76-s − 1.68e5·79-s + 2.06e5·89-s − 2.35e5·101-s − 4.03e5·109-s + 3.90e5·116-s − 1.63e5·121-s − 2.49e5·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 3/2·4-s + 0.0598·11-s + 0.777·16-s − 5.35·19-s + 1.79·29-s − 0.971·31-s − 2.08·41-s + 0.0897·44-s + 1.56·49-s − 4.78·59-s + 0.503·61-s + 0.457·64-s − 4.62·71-s − 8.03·76-s − 3.02·79-s + 2.75·89-s − 2.29·101-s − 3.25·109-s + 2.69·116-s − 1.01·121-s − 1.45·124-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.028323882\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.028323882\) |
| \(L(\frac{7}{2})\) |
|
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 | $D_4\times C_2$ | \( 1 - 3 p^{4} T^{2} + 377 p^{2} T^{4} - 3 p^{14} T^{6} + p^{20} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 26314 T^{2} + 552283323 T^{4} - 26314 p^{10} T^{6} + p^{20} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 12 T + 82074 T^{2} - 12 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 358322 T^{2} - 9568609917 T^{4} - 358322 p^{10} T^{6} + p^{20} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 4107900 T^{2} + 8104643465798 T^{4} - 4107900 p^{10} T^{6} + p^{20} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 4214 T + 8516703 T^{2} + 4214 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 387244 T^{2} + 80324387862918 T^{4} + 387244 p^{10} T^{6} + p^{20} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 4068 T + 45062238 T^{2} - 4068 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 2598 T + 31786727 T^{2} + 2598 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 72084044 T^{2} + 9012846235728918 T^{4} - 72084044 p^{10} T^{6} + p^{20} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 11232 T + 260535762 T^{2} + 11232 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 549439850 T^{2} + 118590729812334123 T^{4} - 549439850 p^{10} T^{6} + p^{20} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 227668020 T^{2} + 116078671176173798 T^{4} - 227668020 p^{10} T^{6} + p^{20} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 447693324 T^{2} + 391810911498356438 T^{4} + 447693324 p^{10} T^{6} + p^{20} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 63924 T + 2170928058 T^{2} + 63924 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 7310 T + 1671053643 T^{2} - 7310 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 1844169850 T^{2} + 1350269600568616923 T^{4} - 1844169850 p^{10} T^{6} + p^{20} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 98304 T + 4832737806 T^{2} + 98304 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 7219366556 T^{2} + 21370998290339231718 T^{4} - 7219366556 p^{10} T^{6} + p^{20} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 84000 T + 7803834398 T^{2} + 84000 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 13556615492 T^{2} + 76898440141363796118 T^{4} - 13556615492 p^{10} T^{6} + p^{20} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 103104 T + 13514604658 T^{2} - 103104 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 1716348290 T^{2} + 75484636759946814723 T^{4} - 1716348290 p^{10} T^{6} + p^{20} T^{8} \) |
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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
−7.997212319232811470846563325933, −7.78773332026863403681360474645, −7.52691229008657996168905689391, −6.95550813802724322679877474405, −6.83687515837145017645282466448, −6.72562627703193363460419509245, −6.53459688508675829401149834849, −6.18065654056461385732907881979, −5.85698969347914744635188421979, −5.83495367100824717923440682158, −5.37731593029619631095594697831, −4.65294735641049262991736488667, −4.51631412683117046807052934164, −4.50927191281712904271722509528, −3.95160731532707225725628868670, −3.89705811419206381469276025779, −2.97462837005634525613566658417, −2.84077997019385980754053810580, −2.83862133278444058377846641477, −2.14592821789794319457558039385, −1.83488104491242319652753227611, −1.65393619119655709272412373829, −1.45871696101823105895353187750, −0.36410018690208853334437990119, −0.29987960369394407494507233874,
0.29987960369394407494507233874, 0.36410018690208853334437990119, 1.45871696101823105895353187750, 1.65393619119655709272412373829, 1.83488104491242319652753227611, 2.14592821789794319457558039385, 2.83862133278444058377846641477, 2.84077997019385980754053810580, 2.97462837005634525613566658417, 3.89705811419206381469276025779, 3.95160731532707225725628868670, 4.50927191281712904271722509528, 4.51631412683117046807052934164, 4.65294735641049262991736488667, 5.37731593029619631095594697831, 5.83495367100824717923440682158, 5.85698969347914744635188421979, 6.18065654056461385732907881979, 6.53459688508675829401149834849, 6.72562627703193363460419509245, 6.83687515837145017645282466448, 6.95550813802724322679877474405, 7.52691229008657996168905689391, 7.78773332026863403681360474645, 7.997212319232811470846563325933
