| L(s) = 1 | − 8·4-s + 36·11-s + 48·16-s − 86·19-s − 60·29-s + 296·31-s + 60·41-s − 288·44-s + 347·49-s + 48·59-s + 1.22e3·61-s − 256·64-s − 312·71-s + 688·76-s + 1.03e3·79-s + 3.07e3·89-s + 4.62e3·101-s − 1.24e3·109-s + 480·116-s − 752·121-s − 2.36e3·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 4-s + 0.986·11-s + 3/4·16-s − 1.03·19-s − 0.384·29-s + 1.71·31-s + 0.228·41-s − 0.986·44-s + 1.01·49-s + 0.105·59-s + 2.57·61-s − 1/2·64-s − 0.521·71-s + 1.03·76-s + 1.47·79-s + 3.65·89-s + 4.55·101-s − 1.09·109-s + 0.384·116-s − 0.564·121-s − 1.71·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(9.096156735\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.096156735\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 347 T^{2} + 185928 T^{4} - 347 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 18 T + 862 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 7823 T^{2} + 24930408 T^{4} - 7823 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 15872 T^{2} + 111187518 T^{4} - 15872 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 43 T + 13710 T^{2} + 43 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 24908 T^{2} + 320075718 T^{4} - 24908 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 30 T + 47122 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 148 T + 63177 T^{2} - 148 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 40730 T^{2} + 1964009643 T^{4} - 40730 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 30 T + 121138 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 133298 T^{2} + 17012554803 T^{4} - 133298 p^{6} T^{6} + p^{12} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 270680 T^{2} + 35119990158 T^{4} - 270680 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 53312 T^{2} - 20927563410 T^{4} - 53312 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 24 T + 380806 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 613 T + 205092 T^{2} - 613 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 236219 T^{2} + 35052520272 T^{4} - 236219 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 156 T + 112462 T^{2} + 156 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 454586 T^{2} + 223496476827 T^{4} - 454586 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $C_2$ | \( ( 1 - 259 T + p^{3} T^{2} )^{4} \) |
| 83 | $D_4\times C_2$ | \( 1 + 1330312 T^{2} + 1088439166878 T^{4} + 1330312 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 1536 T + 1728898 T^{2} - 1536 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 2577503 T^{2} + 3040417246080 T^{4} - 2577503 p^{6} T^{6} + p^{12} T^{8} \) |
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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
−6.60683375506657687998802965180, −6.13536403921087098368642540321, −6.08803064800938601901120605674, −5.88872997032150655150788148442, −5.55944797643248871226212631855, −5.24189010952941178065969078781, −5.13350729104360279913139149863, −5.04236349726292161060256175954, −4.47241038611572096937955165357, −4.37664271466764853060064950404, −4.29751972021202445658724727704, −4.04815639473287658934909401088, −3.87111062130187504674482989423, −3.39279885596293806597417642388, −3.18599374815308771647452241063, −3.16631724973921880594036818343, −2.77200899529916267051519021734, −2.14238144736018084255714003078, −2.09892117671996887515267452955, −1.97205607951920967532243044323, −1.58568832292170214105425241675, −0.980009935474815597807979048272, −0.63780643876065561056703729142, −0.59431040988901861062911327124, −0.54354537560786328102751311026,
0.54354537560786328102751311026, 0.59431040988901861062911327124, 0.63780643876065561056703729142, 0.980009935474815597807979048272, 1.58568832292170214105425241675, 1.97205607951920967532243044323, 2.09892117671996887515267452955, 2.14238144736018084255714003078, 2.77200899529916267051519021734, 3.16631724973921880594036818343, 3.18599374815308771647452241063, 3.39279885596293806597417642388, 3.87111062130187504674482989423, 4.04815639473287658934909401088, 4.29751972021202445658724727704, 4.37664271466764853060064950404, 4.47241038611572096937955165357, 5.04236349726292161060256175954, 5.13350729104360279913139149863, 5.24189010952941178065969078781, 5.55944797643248871226212631855, 5.88872997032150655150788148442, 6.08803064800938601901120605674, 6.13536403921087098368642540321, 6.60683375506657687998802965180
