L(s) = 1 | + 4·5-s − 3·9-s + 16·19-s + 2·25-s + 12·29-s − 8·43-s − 12·45-s − 16·47-s + 49-s + 12·53-s − 8·67-s − 16·71-s + 20·73-s + 9·81-s + 64·95-s − 12·97-s + 4·101-s − 6·121-s − 28·125-s + 127-s + 131-s + 137-s + 139-s + 48·145-s + 149-s + 151-s + 157-s + ⋯ |
L(s) = 1 | + 1.78·5-s − 9-s + 3.67·19-s + 2/5·25-s + 2.22·29-s − 1.21·43-s − 1.78·45-s − 2.33·47-s + 1/7·49-s + 1.64·53-s − 0.977·67-s − 1.89·71-s + 2.34·73-s + 81-s + 6.56·95-s − 1.21·97-s + 0.398·101-s − 0.545·121-s − 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 3.98·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225792 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225792 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.497939845\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.497939845\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−9.280504664575016660349009750311, −8.509354664621903192772158046209, −8.188420417684287628984093773695, −7.58750440596713449607840635902, −7.07204727903132066232126574118, −6.42262084813563967264199691637, −6.11340833663381996666748371567, −5.48790807221177914788155934026, −5.22316409435338364257345412913, −4.86758911716003878330800803137, −3.73233091133641602778161609628, −2.99737772195369908741226640050, −2.79183800612725741835263061431, −1.79239147115953487472752121748, −1.07526755386223660246135627512,
1.07526755386223660246135627512, 1.79239147115953487472752121748, 2.79183800612725741835263061431, 2.99737772195369908741226640050, 3.73233091133641602778161609628, 4.86758911716003878330800803137, 5.22316409435338364257345412913, 5.48790807221177914788155934026, 6.11340833663381996666748371567, 6.42262084813563967264199691637, 7.07204727903132066232126574118, 7.58750440596713449607840635902, 8.188420417684287628984093773695, 8.509354664621903192772158046209, 9.280504664575016660349009750311