| L(s) = 1 | − 7-s − 3·9-s − 4·11-s + 6·13-s − 2·17-s + 6·29-s − 8·31-s + 10·37-s + 2·41-s + 4·43-s + 8·47-s + 49-s + 2·53-s + 8·59-s − 14·61-s + 3·63-s − 12·67-s + 16·71-s − 2·73-s + 4·77-s + 8·79-s + 9·81-s + 8·83-s + 10·89-s − 6·91-s − 2·97-s + 12·99-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 9-s − 1.20·11-s + 1.66·13-s − 0.485·17-s + 1.11·29-s − 1.43·31-s + 1.64·37-s + 0.312·41-s + 0.609·43-s + 1.16·47-s + 1/7·49-s + 0.274·53-s + 1.04·59-s − 1.79·61-s + 0.377·63-s − 1.46·67-s + 1.89·71-s − 0.234·73-s + 0.455·77-s + 0.900·79-s + 81-s + 0.878·83-s + 1.05·89-s − 0.628·91-s − 0.203·97-s + 1.20·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.401898985\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.401898985\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−18.57253169038225, −18.16437397656273, −17.68358075861944, −16.68170845057346, −16.26951127645794, −15.62511391693428, −15.06500081183901, −14.16547912680398, −13.57940150794464, −13.07936770236951, −12.37675015798141, −11.46630999545775, −10.85162413401503, −10.48952017451224, −9.335388238387631, −8.804167587109300, −8.107288137075358, −7.412972465405348, −6.241903473891820, −5.915783810064358, −4.995166840891652, −3.942483550249555, −3.075782092306130, −2.266399988755182, −0.7142897041111533,
0.7142897041111533, 2.266399988755182, 3.075782092306130, 3.942483550249555, 4.995166840891652, 5.915783810064358, 6.241903473891820, 7.412972465405348, 8.107288137075358, 8.804167587109300, 9.335388238387631, 10.48952017451224, 10.85162413401503, 11.46630999545775, 12.37675015798141, 13.07936770236951, 13.57940150794464, 14.16547912680398, 15.06500081183901, 15.62511391693428, 16.26951127645794, 16.68170845057346, 17.68358075861944, 18.16437397656273, 18.57253169038225
