| L(s) = 1 | + 2·3-s − 4·5-s + 2·9-s − 8·15-s − 12·19-s − 8·23-s + 6·27-s + 28·29-s − 20·43-s − 8·45-s + 32·47-s + 16·49-s − 4·53-s − 24·57-s − 12·67-s − 16·69-s − 8·71-s + 8·73-s + 11·81-s + 56·87-s + 48·95-s − 16·97-s − 20·101-s + 32·115-s + 16·121-s + 20·125-s + 127-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.78·5-s + 2/3·9-s − 2.06·15-s − 2.75·19-s − 1.66·23-s + 1.15·27-s + 5.19·29-s − 3.04·43-s − 1.19·45-s + 4.66·47-s + 16/7·49-s − 0.549·53-s − 3.17·57-s − 1.46·67-s − 1.92·69-s − 0.949·71-s + 0.936·73-s + 11/9·81-s + 6.00·87-s + 4.92·95-s − 1.62·97-s − 1.99·101-s + 2.98·115-s + 1.45·121-s + 1.78·125-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.488575444\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.488575444\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| good | 5 | $D_{4}$ | \( ( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 142 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_4\times C_2$ | \( 1 - 16 T^{2} + 126 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 358 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 6 T + 42 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 - 14 T + 102 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 96 T^{2} + 4046 T^{4} - 96 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 2902 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 116 T^{2} + 6406 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 10 T + 106 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 53 | $D_{4}$ | \( ( 1 + 2 T + 102 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 224 T^{2} + 19486 T^{4} - 224 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 172 T^{2} + 13558 T^{4} - 172 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 6 T + 98 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 4 T - 34 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 79 | $D_4\times C_2$ | \( 1 - 128 T^{2} + 7758 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 272 T^{2} + 31774 T^{4} - 272 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 162 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 8 T + 190 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.58466529537121823514154997726, −7.18335137708237561618742043769, −7.08066195669088376767250866621, −6.59335924568600896717028934211, −6.53121168805146507199738683746, −6.47855245082020739774002256760, −6.25763693422057212140713946382, −5.63239728104520187500111972275, −5.55933353659882848251995154607, −5.44525734300318499721014167211, −4.66566724767796312911858904128, −4.59561363840731481462181176044, −4.42968988332501770349683343135, −4.14079054318181423031266963422, −4.12669712169994804630329275532, −3.85816401265561174507361954474, −3.49480344869575337104572687181, −2.94824253369975586331559108812, −2.89963240507423642662031712386, −2.60534742539296043482904941175, −2.39321899657492416575731562878, −1.77009556746866831490971604721, −1.68962093547285754843264970708, −0.68764621073412762469194215300, −0.55743542511046815730854656783,
0.55743542511046815730854656783, 0.68764621073412762469194215300, 1.68962093547285754843264970708, 1.77009556746866831490971604721, 2.39321899657492416575731562878, 2.60534742539296043482904941175, 2.89963240507423642662031712386, 2.94824253369975586331559108812, 3.49480344869575337104572687181, 3.85816401265561174507361954474, 4.12669712169994804630329275532, 4.14079054318181423031266963422, 4.42968988332501770349683343135, 4.59561363840731481462181176044, 4.66566724767796312911858904128, 5.44525734300318499721014167211, 5.55933353659882848251995154607, 5.63239728104520187500111972275, 6.25763693422057212140713946382, 6.47855245082020739774002256760, 6.53121168805146507199738683746, 6.59335924568600896717028934211, 7.08066195669088376767250866621, 7.18335137708237561618742043769, 7.58466529537121823514154997726
