L(s) = 1 | + 2-s − 3-s − 4-s − 2·5-s − 6-s + 4·7-s − 3·8-s + 9-s − 2·10-s + 11-s + 12-s − 2·13-s + 4·14-s + 2·15-s − 16-s − 2·17-s + 18-s + 2·20-s − 4·21-s + 22-s + 8·23-s + 3·24-s − 25-s − 2·26-s − 27-s − 4·28-s − 6·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.894·5-s − 0.408·6-s + 1.51·7-s − 1.06·8-s + 1/3·9-s − 0.632·10-s + 0.301·11-s + 0.288·12-s − 0.554·13-s + 1.06·14-s + 0.516·15-s − 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.447·20-s − 0.872·21-s + 0.213·22-s + 1.66·23-s + 0.612·24-s − 1/5·25-s − 0.392·26-s − 0.192·27-s − 0.755·28-s − 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7473391477\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7473391477\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−16.90261044466837485891274190553, −15.19333100007082269459767774857, −14.59050172699885087838426827684, −13.10139310227399381153384125732, −11.86897475867739185827101909293, −11.06987917382067337680927181024, −8.975261543279013968212899152873, −7.46995816184461358823952634116, −5.31622482563829752699303340190, −4.20407903283329927801590362766,
4.20407903283329927801590362766, 5.31622482563829752699303340190, 7.46995816184461358823952634116, 8.975261543279013968212899152873, 11.06987917382067337680927181024, 11.86897475867739185827101909293, 13.10139310227399381153384125732, 14.59050172699885087838426827684, 15.19333100007082269459767774857, 16.90261044466837485891274190553