L(s) = 1 | + 2·3-s + 9-s + 11-s − 4·13-s − 4·19-s − 4·27-s − 6·29-s − 10·31-s + 2·33-s − 2·37-s − 8·39-s + 12·41-s − 4·43-s − 6·47-s + 6·53-s − 8·57-s − 6·59-s + 4·61-s − 4·67-s − 12·71-s − 4·73-s − 8·79-s − 11·81-s − 12·83-s − 12·87-s − 18·89-s − 20·93-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 1/3·9-s + 0.301·11-s − 1.10·13-s − 0.917·19-s − 0.769·27-s − 1.11·29-s − 1.79·31-s + 0.348·33-s − 0.328·37-s − 1.28·39-s + 1.87·41-s − 0.609·43-s − 0.875·47-s + 0.824·53-s − 1.05·57-s − 0.781·59-s + 0.512·61-s − 0.488·67-s − 1.42·71-s − 0.468·73-s − 0.900·79-s − 1.22·81-s − 1.31·83-s − 1.28·87-s − 1.90·89-s − 2.07·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 215600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 215600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7632737215\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7632737215\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−13.06740922480965, −12.57948336842178, −12.26834469025702, −11.45466937635471, −11.19493623200852, −10.68419625719598, −9.869603774374102, −9.772663957894212, −9.129963455770916, −8.749201143629806, −8.422399489450712, −7.700231662184346, −7.294395720981302, −7.113789466522662, −6.220726552403955, −5.732007081977008, −5.273608005698102, −4.506260282015681, −4.047756746069422, −3.629387160993713, −2.852967548238444, −2.584906336809378, −1.835546756981925, −1.502325018036719, −0.2034200916869148,
0.2034200916869148, 1.502325018036719, 1.835546756981925, 2.584906336809378, 2.852967548238444, 3.629387160993713, 4.047756746069422, 4.506260282015681, 5.273608005698102, 5.732007081977008, 6.220726552403955, 7.113789466522662, 7.294395720981302, 7.700231662184346, 8.422399489450712, 8.749201143629806, 9.129963455770916, 9.772663957894212, 9.869603774374102, 10.68419625719598, 11.19493623200852, 11.45466937635471, 12.26834469025702, 12.57948336842178, 13.06740922480965