L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 2·7-s − 8-s + 9-s + 3·11-s + 12-s + 2·13-s − 2·14-s + 16-s + 6·17-s − 18-s + 19-s + 2·21-s − 3·22-s + 3·23-s − 24-s − 2·26-s + 27-s + 2·28-s + 3·29-s − 7·31-s − 32-s + 3·33-s − 6·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.904·11-s + 0.288·12-s + 0.554·13-s − 0.534·14-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.229·19-s + 0.436·21-s − 0.639·22-s + 0.625·23-s − 0.204·24-s − 0.392·26-s + 0.192·27-s + 0.377·28-s + 0.557·29-s − 1.25·31-s − 0.176·32-s + 0.522·33-s − 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.147156801\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.147156801\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−8.700776189456814343059329719806, −8.181129888235201605412398752993, −7.44187095044134354286726063209, −6.76586273202473007828481468613, −5.79588353287159439386810677108, −4.92255242446553058103900493285, −3.78744051785303354078864018732, −3.09980922754213715117512024374, −1.80044149739624314798440968006, −1.09292154493202384227176417490,
1.09292154493202384227176417490, 1.80044149739624314798440968006, 3.09980922754213715117512024374, 3.78744051785303354078864018732, 4.92255242446553058103900493285, 5.79588353287159439386810677108, 6.76586273202473007828481468613, 7.44187095044134354286726063209, 8.181129888235201605412398752993, 8.700776189456814343059329719806