| L(s) = 1 | + 3-s + 7-s − 2·9-s + 5·11-s − 2·13-s + 5·17-s − 5·19-s + 21-s + 23-s − 5·27-s − 2·29-s + 4·31-s + 5·33-s − 4·37-s − 2·39-s − 5·41-s − 8·43-s − 6·47-s + 49-s + 5·51-s + 4·53-s − 5·57-s − 4·59-s + 10·61-s − 2·63-s + 67-s + 69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s − 2/3·9-s + 1.50·11-s − 0.554·13-s + 1.21·17-s − 1.14·19-s + 0.218·21-s + 0.208·23-s − 0.962·27-s − 0.371·29-s + 0.718·31-s + 0.870·33-s − 0.657·37-s − 0.320·39-s − 0.780·41-s − 1.21·43-s − 0.875·47-s + 1/7·49-s + 0.700·51-s + 0.549·53-s − 0.662·57-s − 0.520·59-s + 1.28·61-s − 0.251·63-s + 0.122·67-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
| \(L(1)\) |
\(\approx\) |
\(2.843617004\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.843617004\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 15 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.18605601663500, −13.96487484059099, −13.34577339894076, −12.67399915192686, −12.11602739229947, −11.79969181503023, −11.28738299094523, −10.73216194062733, −9.966090046263522, −9.678907657366633, −9.055758006523771, −8.444667316967063, −8.271741110800642, −7.593846055686053, −6.828649728890003, −6.539368552087858, −5.796573181214785, −5.197813000011725, −4.648402621937068, −3.805180784643071, −3.520020686031738, −2.777632093997830, −2.014370378702107, −1.495576646970969, −0.5430736501484552,
0.5430736501484552, 1.495576646970969, 2.014370378702107, 2.777632093997830, 3.520020686031738, 3.805180784643071, 4.648402621937068, 5.197813000011725, 5.796573181214785, 6.539368552087858, 6.828649728890003, 7.593846055686053, 8.271741110800642, 8.444667316967063, 9.055758006523771, 9.678907657366633, 9.966090046263522, 10.73216194062733, 11.28738299094523, 11.79969181503023, 12.11602739229947, 12.67399915192686, 13.34577339894076, 13.96487484059099, 14.18605601663500
