L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 7-s − 8-s + 9-s − 11-s − 12-s − 2·13-s − 14-s + 16-s + 2·17-s − 18-s + 4·19-s − 21-s + 22-s − 4·23-s + 24-s + 2·26-s − 27-s + 28-s + 6·29-s − 32-s + 33-s − 2·34-s + 36-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s − 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.917·19-s − 0.218·21-s + 0.213·22-s − 0.834·23-s + 0.204·24-s + 0.392·26-s − 0.192·27-s + 0.188·28-s + 1.11·29-s − 0.176·32-s + 0.174·33-s − 0.342·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.146401693\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.146401693\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−16.52412186554046, −15.99399090632721, −15.51281135284695, −14.77282448794228, −14.25730752794809, −13.61815914191578, −12.85211954513692, −12.15042428091574, −11.86412257243516, −11.18408299088954, −10.61043879787277, −9.960076961606691, −9.626830690072863, −8.804085361057445, −8.061626021316880, −7.582033928208297, −7.059153188006784, −6.132067695442210, −5.709347561370897, −4.873583254679838, −4.257382335983982, −3.168679824117194, −2.430352058312982, −1.443034345718492, −0.6001011691945772,
0.6001011691945772, 1.443034345718492, 2.430352058312982, 3.168679824117194, 4.257382335983982, 4.873583254679838, 5.709347561370897, 6.132067695442210, 7.059153188006784, 7.582033928208297, 8.061626021316880, 8.804085361057445, 9.626830690072863, 9.960076961606691, 10.61043879787277, 11.18408299088954, 11.86412257243516, 12.15042428091574, 12.85211954513692, 13.61815914191578, 14.25730752794809, 14.77282448794228, 15.51281135284695, 15.99399090632721, 16.52412186554046