L(s) = 1 | − 2·5-s + 8·13-s − 4·17-s + 3·25-s + 12·29-s + 16·37-s − 16·41-s + 6·49-s + 12·53-s + 20·61-s − 16·65-s + 12·73-s + 8·85-s − 8·89-s + 4·97-s − 20·101-s − 12·109-s − 28·113-s − 2·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s − 24·145-s + 149-s + 151-s + ⋯ |
L(s) = 1 | − 0.894·5-s + 2.21·13-s − 0.970·17-s + 3/5·25-s + 2.22·29-s + 2.63·37-s − 2.49·41-s + 6/7·49-s + 1.64·53-s + 2.56·61-s − 1.98·65-s + 1.40·73-s + 0.867·85-s − 0.847·89-s + 0.406·97-s − 1.99·101-s − 1.14·109-s − 2.63·113-s − 0.181·121-s − 0.357·125-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.243947625\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.243947625\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.599542367778113449539114677714, −9.373791719654431270600819972011, −8.654771683914173419698468705002, −8.495905321502352456562376203856, −8.166052791193086010050666688408, −8.064085360155866759608363816890, −7.16226301259578703183065798768, −6.85326970006485740939466684428, −6.48775262574578907813008021114, −6.26337537079386845543709924020, −5.41073814013941971526674776398, −5.34944635106809400960610371164, −4.41054170800755004985245042305, −4.20678341357415009135671802046, −3.86396293560103417260816712209, −3.23572458576821090731290642935, −2.75396112564473314256842278454, −2.12132454984846043350366901398, −1.16402186534327768941641869551, −0.72225113761563103121873608148,
0.72225113761563103121873608148, 1.16402186534327768941641869551, 2.12132454984846043350366901398, 2.75396112564473314256842278454, 3.23572458576821090731290642935, 3.86396293560103417260816712209, 4.20678341357415009135671802046, 4.41054170800755004985245042305, 5.34944635106809400960610371164, 5.41073814013941971526674776398, 6.26337537079386845543709924020, 6.48775262574578907813008021114, 6.85326970006485740939466684428, 7.16226301259578703183065798768, 8.064085360155866759608363816890, 8.166052791193086010050666688408, 8.495905321502352456562376203856, 8.654771683914173419698468705002, 9.373791719654431270600819972011, 9.599542367778113449539114677714