L(s) = 1 | − 2-s + 3-s + 4-s − 3·5-s − 6-s + 5·7-s − 8-s + 9-s + 3·10-s + 3·11-s + 12-s + 2·13-s − 5·14-s − 3·15-s + 16-s − 3·17-s − 18-s − 4·19-s − 3·20-s + 5·21-s − 3·22-s − 3·23-s − 24-s + 4·25-s − 2·26-s + 27-s + 5·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s − 0.408·6-s + 1.88·7-s − 0.353·8-s + 1/3·9-s + 0.948·10-s + 0.904·11-s + 0.288·12-s + 0.554·13-s − 1.33·14-s − 0.774·15-s + 1/4·16-s − 0.727·17-s − 0.235·18-s − 0.917·19-s − 0.670·20-s + 1.09·21-s − 0.639·22-s − 0.625·23-s − 0.204·24-s + 4/5·25-s − 0.392·26-s + 0.192·27-s + 0.944·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 786 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 786 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.392005187\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.392005187\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 131 | \( 1 + T \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−10.52012305463675061890377086278, −8.912370674872702733722534445204, −8.669967477699422663489616704290, −7.82822504919717217718631672319, −7.33290336080518283375732110140, −6.09216702551084552410180646079, −4.41337572182358430917124977668, −4.08712085170387909418505833895, −2.41205303586958044192237087396, −1.15147573070191873017096990848,
1.15147573070191873017096990848, 2.41205303586958044192237087396, 4.08712085170387909418505833895, 4.41337572182358430917124977668, 6.09216702551084552410180646079, 7.33290336080518283375732110140, 7.82822504919717217718631672319, 8.669967477699422663489616704290, 8.912370674872702733722534445204, 10.52012305463675061890377086278