| L(s) = 1 | − 3-s − 7-s − 2·9-s + 2·11-s − 3·13-s + 6·17-s − 6·19-s + 21-s + 23-s + 5·27-s + 5·29-s + 7·31-s − 2·33-s + 4·37-s + 3·39-s − 3·41-s + 10·43-s + 7·47-s + 49-s − 6·51-s + 4·53-s + 6·57-s + 4·59-s + 12·61-s + 2·63-s − 4·67-s − 69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s − 2/3·9-s + 0.603·11-s − 0.832·13-s + 1.45·17-s − 1.37·19-s + 0.218·21-s + 0.208·23-s + 0.962·27-s + 0.928·29-s + 1.25·31-s − 0.348·33-s + 0.657·37-s + 0.480·39-s − 0.468·41-s + 1.52·43-s + 1.02·47-s + 1/7·49-s − 0.840·51-s + 0.549·53-s + 0.794·57-s + 0.520·59-s + 1.53·61-s + 0.251·63-s − 0.488·67-s − 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.924265135\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.924265135\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−14.36106332969162, −13.76825452876698, −13.22181294889679, −12.44453433511443, −12.19153233221025, −11.91957640802799, −11.21087316483452, −10.62382421609415, −10.29249571428206, −9.636721260486374, −9.244893936738248, −8.404581945294540, −8.230248377372289, −7.393284377027603, −6.857288923357662, −6.258288155249132, −5.936096539283446, −5.250012390360955, −4.722468493635462, −4.067837388584774, −3.431236581827145, −2.632033135617718, −2.271115542733046, −1.001975824347975, −0.5985349955513300,
0.5985349955513300, 1.001975824347975, 2.271115542733046, 2.632033135617718, 3.431236581827145, 4.067837388584774, 4.722468493635462, 5.250012390360955, 5.936096539283446, 6.258288155249132, 6.857288923357662, 7.393284377027603, 8.230248377372289, 8.404581945294540, 9.244893936738248, 9.636721260486374, 10.29249571428206, 10.62382421609415, 11.21087316483452, 11.91957640802799, 12.19153233221025, 12.44453433511443, 13.22181294889679, 13.76825452876698, 14.36106332969162
