| L(s) = 1 | + 16·16-s + 96·67-s − 96·101-s + 72·103-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 100·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯ |
| L(s) = 1 | + 4·16-s + 11.7·67-s − 9.55·101-s + 7.09·103-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 7.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.04959194\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.04959194\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T^{8} \) |
| 17 | \( 1 \) |
| good | 2 | \( ( 1 - p T^{2} )^{4}( 1 + p T^{2} )^{4} \) |
| 5 | \( 1 + 1151 T^{8} + p^{8} T^{16} \) |
| 7 | \( 1 - 4798 T^{8} + p^{8} T^{16} \) |
| 11 | \( 1 + 25007 T^{8} + p^{8} T^{16} \) |
| 13 | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - 553 T^{4} + p^{4} T^{8} )^{2} \) |
| 23 | \( 1 + 375407 T^{8} + p^{8} T^{16} \) |
| 29 | \( 1 + 20642 T^{8} + p^{8} T^{16} \) |
| 31 | \( 1 - 1618558 T^{8} + p^{8} T^{16} \) |
| 37 | \( ( 1 - 1680 T^{4} + p^{4} T^{8} )( 1 + 1680 T^{4} + p^{4} T^{8} ) \) |
| 41 | \( 1 - 5638753 T^{8} + p^{8} T^{16} \) |
| 43 | \( ( 1 + 3527 T^{4} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 90 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 - 718 T^{4} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 2640 T^{4} + p^{4} T^{8} )( 1 + 2640 T^{4} + p^{4} T^{8} ) \) |
| 67 | \( ( 1 - 12 T + p T^{2} )^{8} \) |
| 71 | \( ( 1 + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 10560 T^{4} + p^{4} T^{8} )( 1 + 10560 T^{4} + p^{4} T^{8} ) \) |
| 79 | \( 1 - 18887038 T^{8} + p^{8} T^{16} \) |
| 83 | \( ( 1 + 3122 T^{4} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 16 T + p T^{2} )^{4}( 1 + 16 T + p T^{2} )^{4} \) |
| 97 | \( ( 1 - 18720 T^{4} + p^{4} T^{8} )( 1 + 18720 T^{4} + p^{4} T^{8} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.38803365260635213850301115801, −4.23647695548400106968917202909, −4.17062337058418735897085056584, −3.88464749561289881567933701158, −3.65253261344827471459674901570, −3.63079220830614793736056521422, −3.54387693848652329460525420976, −3.44165469233855412519132552047, −3.42697979692763419976340983876, −3.32641025043870070519595329158, −3.06473020925539017408501389026, −2.87153264331061141918570812228, −2.70628379062639720512705682550, −2.42625859692382689967906667081, −2.35666852036228696635976781442, −2.22482619869954410039862246944, −2.11605467386941435410205557479, −1.89636048052674727802589319131, −1.73129276358341186624772577399, −1.42177737747814760269223061195, −1.13598270452360027363626434685, −1.01186030731210054555484842785, −0.935662553143527501138593862503, −0.61950327496178196801726476827, −0.45580765796521759949327945374,
0.45580765796521759949327945374, 0.61950327496178196801726476827, 0.935662553143527501138593862503, 1.01186030731210054555484842785, 1.13598270452360027363626434685, 1.42177737747814760269223061195, 1.73129276358341186624772577399, 1.89636048052674727802589319131, 2.11605467386941435410205557479, 2.22482619869954410039862246944, 2.35666852036228696635976781442, 2.42625859692382689967906667081, 2.70628379062639720512705682550, 2.87153264331061141918570812228, 3.06473020925539017408501389026, 3.32641025043870070519595329158, 3.42697979692763419976340983876, 3.44165469233855412519132552047, 3.54387693848652329460525420976, 3.63079220830614793736056521422, 3.65253261344827471459674901570, 3.88464749561289881567933701158, 4.17062337058418735897085056584, 4.23647695548400106968917202909, 4.38803365260635213850301115801
Plot not available for L-functions of degree greater than 10.
