| L(s) = 1 | − 0.465·3-s − 2.78·9-s + 1.37·11-s − 5.12·13-s − 0.337·17-s + 5.99·19-s + 6.54·23-s + 2.69·27-s − 7.99·29-s − 7.24·31-s − 0.641·33-s + 6.05·37-s + 2.38·39-s + 6.68·41-s − 5.02·43-s + 6.64·47-s + 0.157·51-s + 4.87·53-s − 2.79·57-s + 0.602·59-s − 13.0·61-s + 12.1·67-s − 3.05·69-s − 1.39·71-s − 1.68·73-s + 7.61·79-s + 7.09·81-s + ⋯ |
| L(s) = 1 | − 0.268·3-s − 0.927·9-s + 0.415·11-s − 1.42·13-s − 0.0819·17-s + 1.37·19-s + 1.36·23-s + 0.518·27-s − 1.48·29-s − 1.30·31-s − 0.111·33-s + 0.995·37-s + 0.381·39-s + 1.04·41-s − 0.766·43-s + 0.969·47-s + 0.0220·51-s + 0.669·53-s − 0.369·57-s + 0.0784·59-s − 1.67·61-s + 1.47·67-s − 0.367·69-s − 0.165·71-s − 0.197·73-s + 0.856·79-s + 0.788·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 + 0.465T + 3T^{2} \) |
| 11 | \( 1 - 1.37T + 11T^{2} \) |
| 13 | \( 1 + 5.12T + 13T^{2} \) |
| 17 | \( 1 + 0.337T + 17T^{2} \) |
| 19 | \( 1 - 5.99T + 19T^{2} \) |
| 23 | \( 1 - 6.54T + 23T^{2} \) |
| 29 | \( 1 + 7.99T + 29T^{2} \) |
| 31 | \( 1 + 7.24T + 31T^{2} \) |
| 37 | \( 1 - 6.05T + 37T^{2} \) |
| 41 | \( 1 - 6.68T + 41T^{2} \) |
| 43 | \( 1 + 5.02T + 43T^{2} \) |
| 47 | \( 1 - 6.64T + 47T^{2} \) |
| 53 | \( 1 - 4.87T + 53T^{2} \) |
| 59 | \( 1 - 0.602T + 59T^{2} \) |
| 61 | \( 1 + 13.0T + 61T^{2} \) |
| 67 | \( 1 - 12.1T + 67T^{2} \) |
| 71 | \( 1 + 1.39T + 71T^{2} \) |
| 73 | \( 1 + 1.68T + 73T^{2} \) |
| 79 | \( 1 - 7.61T + 79T^{2} \) |
| 83 | \( 1 + 7.49T + 83T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 - 0.691T + 97T^{2} \) |
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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
−7.37545780748262686172155044176, −6.75873598510077586621842631445, −5.74716825741774236611926300268, −5.39368836323551537215618451088, −4.71270748078539997608637002932, −3.75123870857835791490282654837, −2.97238662078296157710582969302, −2.28716685702112908892413095747, −1.10487374219309707579593241986, 0,
1.10487374219309707579593241986, 2.28716685702112908892413095747, 2.97238662078296157710582969302, 3.75123870857835791490282654837, 4.71270748078539997608637002932, 5.39368836323551537215618451088, 5.74716825741774236611926300268, 6.75873598510077586621842631445, 7.37545780748262686172155044176
