| L(s) = 1 | − 3-s + 2.12·5-s + 0.221·7-s + 9-s − 2.12·11-s − 2.07·13-s − 2.12·15-s − 5.53·17-s − 2.18·19-s − 0.221·21-s + 8.50·23-s − 0.481·25-s − 27-s − 9.67·29-s − 3.18·31-s + 2.12·33-s + 0.471·35-s − 5.07·37-s + 2.07·39-s − 3.43·41-s + 5.53·43-s + 2.12·45-s − 11.8·47-s − 6.95·49-s + 5.53·51-s + 2.21·53-s − 4.51·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.950·5-s + 0.0837·7-s + 0.333·9-s − 0.640·11-s − 0.576·13-s − 0.548·15-s − 1.34·17-s − 0.500·19-s − 0.0483·21-s + 1.77·23-s − 0.0963·25-s − 0.192·27-s − 1.79·29-s − 0.572·31-s + 0.370·33-s + 0.0796·35-s − 0.835·37-s + 0.332·39-s − 0.537·41-s + 0.843·43-s + 0.316·45-s − 1.72·47-s − 0.992·49-s + 0.775·51-s + 0.304·53-s − 0.609·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 - 2.12T + 5T^{2} \) |
| 7 | \( 1 - 0.221T + 7T^{2} \) |
| 11 | \( 1 + 2.12T + 11T^{2} \) |
| 13 | \( 1 + 2.07T + 13T^{2} \) |
| 17 | \( 1 + 5.53T + 17T^{2} \) |
| 19 | \( 1 + 2.18T + 19T^{2} \) |
| 23 | \( 1 - 8.50T + 23T^{2} \) |
| 29 | \( 1 + 9.67T + 29T^{2} \) |
| 31 | \( 1 + 3.18T + 31T^{2} \) |
| 37 | \( 1 + 5.07T + 37T^{2} \) |
| 41 | \( 1 + 3.43T + 41T^{2} \) |
| 43 | \( 1 - 5.53T + 43T^{2} \) |
| 47 | \( 1 + 11.8T + 47T^{2} \) |
| 53 | \( 1 - 2.21T + 53T^{2} \) |
| 59 | \( 1 - 6.60T + 59T^{2} \) |
| 61 | \( 1 - 9.72T + 61T^{2} \) |
| 67 | \( 1 + 0.214T + 67T^{2} \) |
| 71 | \( 1 - 12.8T + 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 - 7.45T + 79T^{2} \) |
| 83 | \( 1 - 15.9T + 83T^{2} \) |
| 89 | \( 1 + 6.46T + 89T^{2} \) |
| 97 | \( 1 + 0.984T + 97T^{2} \) |
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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
−8.995249070984132192434659715428, −7.939380270491353117395634737280, −6.96986829070835972101890678517, −6.47484391803185917712916703790, −5.31678799946558743224721415094, −5.10068349644640561394505118954, −3.86231677524924331169908235058, −2.53911303950147513771068285833, −1.71639441740853883944837131569, 0,
1.71639441740853883944837131569, 2.53911303950147513771068285833, 3.86231677524924331169908235058, 5.10068349644640561394505118954, 5.31678799946558743224721415094, 6.47484391803185917712916703790, 6.96986829070835972101890678517, 7.939380270491353117395634737280, 8.995249070984132192434659715428
