| L(s) = 1 | + 3-s − 4-s + 2·5-s + 2·7-s − 12-s + 2·15-s + 16-s − 2·17-s + 8·19-s − 2·20-s + 2·21-s + 2·23-s − 25-s − 4·27-s − 2·28-s + 4·35-s + 8·37-s + 48-s − 2·49-s − 2·51-s + 8·57-s − 4·59-s − 2·60-s − 64-s + 2·68-s + 2·69-s + 8·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1/2·4-s + 0.894·5-s + 0.755·7-s − 0.288·12-s + 0.516·15-s + 1/4·16-s − 0.485·17-s + 1.83·19-s − 0.447·20-s + 0.436·21-s + 0.417·23-s − 1/5·25-s − 0.769·27-s − 0.377·28-s + 0.676·35-s + 1.31·37-s + 0.144·48-s − 2/7·49-s − 0.280·51-s + 1.05·57-s − 0.520·59-s − 0.258·60-s − 1/8·64-s + 0.242·68-s + 0.240·69-s + 0.936·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86700 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86700 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.108638445\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.108638445\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 17 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 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
−9.515787212071587608976987326879, −9.286786413609602707355343813121, −8.814094502953020134316548756674, −8.121545273000048326248326912994, −7.80845566987432003768565197912, −7.32211078113343980519630660530, −6.61631553829504663083294799259, −5.93148042739259581781700849751, −5.47610983754170823461728105342, −4.95535452098970716543893127712, −4.35869357077627461302818756248, −3.58702899448501549862735144331, −2.88264402882531541396188854345, −2.10707937231367077654694312747, −1.22256098708743177778818610952,
1.22256098708743177778818610952, 2.10707937231367077654694312747, 2.88264402882531541396188854345, 3.58702899448501549862735144331, 4.35869357077627461302818756248, 4.95535452098970716543893127712, 5.47610983754170823461728105342, 5.93148042739259581781700849751, 6.61631553829504663083294799259, 7.32211078113343980519630660530, 7.80845566987432003768565197912, 8.121545273000048326248326912994, 8.814094502953020134316548756674, 9.286786413609602707355343813121, 9.515787212071587608976987326879
