| L(s) = 1 | + 2·5-s − 8·11-s + 5·25-s + 12·29-s − 8·31-s − 20·41-s + 2·49-s − 16·55-s − 8·59-s − 4·61-s − 24·79-s − 40·89-s − 4·101-s + 8·109-s + 38·121-s + 22·125-s + 127-s + 131-s + 137-s + 139-s + 24·145-s + 149-s + 151-s − 16·155-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 2.41·11-s + 25-s + 2.22·29-s − 1.43·31-s − 3.12·41-s + 2/7·49-s − 2.15·55-s − 1.04·59-s − 0.512·61-s − 2.70·79-s − 4.23·89-s − 0.398·101-s + 0.766·109-s + 3.45·121-s + 1.96·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.99·145-s + 0.0819·149-s + 0.0813·151-s − 1.28·155-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.593145282\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.593145282\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| good | 7 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )( 1 + 11 T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^2$ | \( ( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 + 30 T^{2} + 371 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 10 T + 59 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 + 70 T^{2} + 3051 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^3$ | \( 1 + 78 T^{2} + 3875 T^{4} + 78 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 118 T^{2} + 9435 T^{4} + 118 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 12 T + 65 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 150 T^{2} + 15611 T^{4} + 150 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 97 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 18 T + 227 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 227 T^{2} + 18 p T^{3} + p^{2} T^{4} ) \) |
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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
−6.87487978240398225533027097127, −6.49463401861219235426859817927, −6.20003609871739595985991089797, −5.95199129584608784783911981574, −5.77033422198008329826933739281, −5.59707034037763628493509348934, −5.51990994717763990302479214893, −5.11432066579614905204482538583, −4.94449793216476995036249330310, −4.84126428978759674506143910746, −4.68091991888395313574868081305, −4.17498207615341211481834169257, −4.14977858283360067024067289776, −3.88041898143849012433281129860, −3.20451743833591257856003084214, −3.12058814302853418786394880592, −2.97917366424530887607066759852, −2.90392644529101305915713833803, −2.56081291717534120858301871847, −2.12720526480300094634206738632, −1.80188548908605471662246378502, −1.61047896617344645290791139035, −1.43599939614480679761744464924, −0.48792051881301496384766339888, −0.47986596674441648582421539474,
0.47986596674441648582421539474, 0.48792051881301496384766339888, 1.43599939614480679761744464924, 1.61047896617344645290791139035, 1.80188548908605471662246378502, 2.12720526480300094634206738632, 2.56081291717534120858301871847, 2.90392644529101305915713833803, 2.97917366424530887607066759852, 3.12058814302853418786394880592, 3.20451743833591257856003084214, 3.88041898143849012433281129860, 4.14977858283360067024067289776, 4.17498207615341211481834169257, 4.68091991888395313574868081305, 4.84126428978759674506143910746, 4.94449793216476995036249330310, 5.11432066579614905204482538583, 5.51990994717763990302479214893, 5.59707034037763628493509348934, 5.77033422198008329826933739281, 5.95199129584608784783911981574, 6.20003609871739595985991089797, 6.49463401861219235426859817927, 6.87487978240398225533027097127
