L(s) = 1 | − 2-s + 3-s − 4-s − 2·5-s − 6-s − 7-s + 3·8-s + 9-s + 2·10-s + 4·11-s − 12-s − 2·13-s + 14-s − 2·15-s − 16-s − 6·17-s − 18-s + 4·19-s + 2·20-s − 21-s − 4·22-s + 3·24-s − 25-s + 2·26-s + 27-s + 28-s − 2·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.894·5-s − 0.408·6-s − 0.377·7-s + 1.06·8-s + 1/3·9-s + 0.632·10-s + 1.20·11-s − 0.288·12-s − 0.554·13-s + 0.267·14-s − 0.516·15-s − 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.917·19-s + 0.447·20-s − 0.218·21-s − 0.852·22-s + 0.612·24-s − 1/5·25-s + 0.392·26-s + 0.192·27-s + 0.188·28-s − 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4511154053\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4511154053\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−18.41922826796404611800756026796, −17.15982418199832408435897093924, −15.87345980006494599806967761956, −14.51353234037878627029755764830, −13.19379250527463570089741680145, −11.55595081754559805664353277943, −9.724661210726173973340366981146, −8.672365196683144779961098572501, −7.25047783802838427330028450541, −4.13559084050773741974089362056,
4.13559084050773741974089362056, 7.25047783802838427330028450541, 8.672365196683144779961098572501, 9.724661210726173973340366981146, 11.55595081754559805664353277943, 13.19379250527463570089741680145, 14.51353234037878627029755764830, 15.87345980006494599806967761956, 17.15982418199832408435897093924, 18.41922826796404611800756026796