L(s) = 1 | − 2-s + 4-s + 7-s − 8-s − 4·11-s + 13-s − 14-s + 16-s + 17-s + 5·19-s + 4·22-s − 4·23-s − 26-s + 28-s + 3·29-s − 8·31-s − 32-s − 34-s + 4·37-s − 5·38-s − 2·41-s − 4·44-s + 4·46-s − 7·47-s + 49-s + 52-s + 9·53-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s − 1.20·11-s + 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.242·17-s + 1.14·19-s + 0.852·22-s − 0.834·23-s − 0.196·26-s + 0.188·28-s + 0.557·29-s − 1.43·31-s − 0.176·32-s − 0.171·34-s + 0.657·37-s − 0.811·38-s − 0.312·41-s − 0.603·44-s + 0.589·46-s − 1.02·47-s + 1/7·49-s + 0.138·52-s + 1.23·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−7.50367641109981775966290977777, −6.92180362547999547092232348644, −5.90880421855443400700146364292, −5.45870560402921971097968161629, −4.68354499860330806921080195770, −3.66920848748091718771719385180, −2.89926524824067951467130963405, −2.07760221439917293574244532825, −1.16377604698832876980348557644, 0,
1.16377604698832876980348557644, 2.07760221439917293574244532825, 2.89926524824067951467130963405, 3.66920848748091718771719385180, 4.68354499860330806921080195770, 5.45870560402921971097968161629, 5.90880421855443400700146364292, 6.92180362547999547092232348644, 7.50367641109981775966290977777