L(s) = 1 | + 2·3-s + 7-s + 9-s − 4·11-s − 6·17-s + 6·19-s + 2·21-s − 23-s − 4·27-s + 10·29-s − 4·31-s − 8·33-s + 2·37-s − 10·41-s − 4·43-s + 12·47-s + 49-s − 12·51-s + 6·53-s + 12·57-s + 2·59-s + 63-s − 2·69-s + 8·71-s + 6·73-s − 4·77-s + 8·79-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 0.377·7-s + 1/3·9-s − 1.20·11-s − 1.45·17-s + 1.37·19-s + 0.436·21-s − 0.208·23-s − 0.769·27-s + 1.85·29-s − 0.718·31-s − 1.39·33-s + 0.328·37-s − 1.56·41-s − 0.609·43-s + 1.75·47-s + 1/7·49-s − 1.68·51-s + 0.824·53-s + 1.58·57-s + 0.260·59-s + 0.125·63-s − 0.240·69-s + 0.949·71-s + 0.702·73-s − 0.455·77-s + 0.900·79-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−14.26283091449530, −13.97255182190155, −13.60507818102044, −13.19503597055374, −12.57982587154416, −11.96025389868892, −11.46316912443489, −10.90962620074717, −10.33247460731786, −9.894921477186559, −9.260060287176258, −8.767179433951714, −8.224880154751960, −8.055820200726789, −7.168857068487436, −6.994495845627769, −6.062450179452018, −5.378294968931349, −4.957508676568268, −4.267655146346289, −3.611224922021197, −2.940087938999464, −2.484947801305722, −1.977587855696725, −1.023776765101948, 0,
1.023776765101948, 1.977587855696725, 2.484947801305722, 2.940087938999464, 3.611224922021197, 4.267655146346289, 4.957508676568268, 5.378294968931349, 6.062450179452018, 6.994495845627769, 7.168857068487436, 8.055820200726789, 8.224880154751960, 8.767179433951714, 9.260060287176258, 9.894921477186559, 10.33247460731786, 10.90962620074717, 11.46316912443489, 11.96025389868892, 12.57982587154416, 13.19503597055374, 13.60507818102044, 13.97255182190155, 14.26283091449530