| L(s) = 1 | − 8·5-s + 8·13-s + 32·25-s − 8·29-s + 40·37-s + 24·49-s + 8·53-s + 40·61-s − 64·65-s + 12·81-s + 64·97-s + 40·101-s − 40·109-s + 16·113-s − 88·125-s + 127-s + 131-s + 137-s + 139-s + 64·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 32·169-s + 173-s + ⋯ |
| L(s) = 1 | − 3.57·5-s + 2.21·13-s + 32/5·25-s − 1.48·29-s + 6.57·37-s + 24/7·49-s + 1.09·53-s + 5.12·61-s − 7.93·65-s + 4/3·81-s + 6.49·97-s + 3.98·101-s − 3.83·109-s + 1.50·113-s − 7.87·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5.31·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.46·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.265107670\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.265107670\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 4 p T^{4} + 70 T^{8} - 4 p^{5} T^{12} + p^{8} T^{16} \) |
| 5 | \( ( 1 + 4 T + 8 T^{2} + 12 T^{3} + 14 T^{4} + 12 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 - 12 T^{2} + 102 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( 1 - 460 T^{4} + 82054 T^{8} - 460 p^{4} T^{12} + p^{8} T^{16} \) |
| 13 | \( ( 1 - 4 T + 8 T^{2} - 44 T^{3} + 238 T^{4} - 44 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( 1 - 12 T^{4} - 247354 T^{8} - 12 p^{4} T^{12} + p^{8} T^{16} \) |
| 23 | \( ( 1 - 12 T^{2} + 1062 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 4 T + 8 T^{2} - 20 T^{3} - 1106 T^{4} - 20 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 60 T^{2} + 2310 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 20 T + 200 T^{2} - 1660 T^{3} + 11662 T^{4} - 1660 p T^{5} + 200 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 66 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( 1 + 6068 T^{4} + 15827590 T^{8} + 6068 p^{4} T^{12} + p^{8} T^{16} \) |
| 47 | \( ( 1 + 124 T^{2} + 7750 T^{4} + 124 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 4 T + 8 T^{2} - 76 T^{3} - 434 T^{4} - 76 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 + 12084 T^{4} + 60324614 T^{8} + 12084 p^{4} T^{12} + p^{8} T^{16} \) |
| 61 | \( ( 1 - 20 T + 200 T^{2} - 1500 T^{3} + 11054 T^{4} - 1500 p T^{5} + 200 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 + 116 T^{4} - 21877946 T^{8} + 116 p^{4} T^{12} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 76 T^{2} + 9958 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 140 T^{2} + 14406 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 60 T^{2} + 5190 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 - 3468 T^{4} - 36032314 T^{8} - 3468 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 332 T^{2} + 43270 T^{4} - 332 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 16 T + 250 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{4} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.80277027601451737875561888981, −4.70408731845672587016690169169, −4.42179928323541649983103938674, −4.14272442028111251683753267287, −4.03168973381028160062146801999, −4.00889399982428431738996244101, −3.95866571589529200275521969013, −3.91982951127300207157823204613, −3.74053036532520893199862754555, −3.51936618390696357762933091420, −3.45252000573128521248805096025, −3.30409461289165851670123381540, −3.24651318719033825981817599854, −2.69082056999746400524263862751, −2.65771095148498799291654201971, −2.55091612815288610558045148441, −2.40696501287321243965024856278, −2.15005906629136943736267029243, −2.13394192043450408939124189694, −1.68858570190644246044479542165, −1.20853224460657712950663456614, −1.00125856704785019149684505453, −0.77224484747695618626007886078, −0.75073934912654972544679828961, −0.53312798155045654810628430163,
0.53312798155045654810628430163, 0.75073934912654972544679828961, 0.77224484747695618626007886078, 1.00125856704785019149684505453, 1.20853224460657712950663456614, 1.68858570190644246044479542165, 2.13394192043450408939124189694, 2.15005906629136943736267029243, 2.40696501287321243965024856278, 2.55091612815288610558045148441, 2.65771095148498799291654201971, 2.69082056999746400524263862751, 3.24651318719033825981817599854, 3.30409461289165851670123381540, 3.45252000573128521248805096025, 3.51936618390696357762933091420, 3.74053036532520893199862754555, 3.91982951127300207157823204613, 3.95866571589529200275521969013, 4.00889399982428431738996244101, 4.03168973381028160062146801999, 4.14272442028111251683753267287, 4.42179928323541649983103938674, 4.70408731845672587016690169169, 4.80277027601451737875561888981
Plot not available for L-functions of degree greater than 10.
