| L(s) = 1 | − 2-s + 4-s + 5-s − 8-s − 10-s + 11-s + 2·13-s + 16-s − 5·17-s − 6·19-s + 20-s − 22-s − 7·23-s − 4·25-s − 2·26-s + 8·29-s + 10·31-s − 32-s + 5·34-s − 8·37-s + 6·38-s − 40-s − 7·41-s + 4·43-s + 44-s + 7·46-s + 47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s + 0.301·11-s + 0.554·13-s + 1/4·16-s − 1.21·17-s − 1.37·19-s + 0.223·20-s − 0.213·22-s − 1.45·23-s − 4/5·25-s − 0.392·26-s + 1.48·29-s + 1.79·31-s − 0.176·32-s + 0.857·34-s − 1.31·37-s + 0.973·38-s − 0.158·40-s − 1.09·41-s + 0.609·43-s + 0.150·44-s + 1.03·46-s + 0.145·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.258869340\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.258869340\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−7.961447425306485333392966125182, −6.81388394122217528415396104787, −6.39025663690856989553309758239, −6.01741760327031268275827234772, −4.84685398339304866072960619184, −4.22514070706333280365812832139, −3.33841410974282725521488441665, −2.24274847033046599353683453732, −1.83793605265318764680664972367, −0.57864134479687790735385258032,
0.57864134479687790735385258032, 1.83793605265318764680664972367, 2.24274847033046599353683453732, 3.33841410974282725521488441665, 4.22514070706333280365812832139, 4.84685398339304866072960619184, 6.01741760327031268275827234772, 6.39025663690856989553309758239, 6.81388394122217528415396104787, 7.961447425306485333392966125182
