| L(s) = 1 | + 0.584·2-s + 3-s − 1.65·4-s + 1.95·5-s + 0.584·6-s − 1.51·7-s − 2.13·8-s + 9-s + 1.14·10-s − 11-s − 1.65·12-s − 0.883·14-s + 1.95·15-s + 2.06·16-s + 3.44·17-s + 0.584·18-s − 5.09·19-s − 3.23·20-s − 1.51·21-s − 0.584·22-s − 0.701·23-s − 2.13·24-s − 1.18·25-s + 27-s + 2.50·28-s + 5.04·29-s + 1.14·30-s + ⋯ |
| L(s) = 1 | + 0.413·2-s + 0.577·3-s − 0.829·4-s + 0.873·5-s + 0.238·6-s − 0.571·7-s − 0.756·8-s + 0.333·9-s + 0.361·10-s − 0.301·11-s − 0.478·12-s − 0.236·14-s + 0.504·15-s + 0.516·16-s + 0.834·17-s + 0.137·18-s − 1.16·19-s − 0.724·20-s − 0.329·21-s − 0.124·22-s − 0.146·23-s − 0.436·24-s − 0.236·25-s + 0.192·27-s + 0.473·28-s + 0.937·29-s + 0.208·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.584T + 2T^{2} \) |
| 5 | \( 1 - 1.95T + 5T^{2} \) |
| 7 | \( 1 + 1.51T + 7T^{2} \) |
| 17 | \( 1 - 3.44T + 17T^{2} \) |
| 19 | \( 1 + 5.09T + 19T^{2} \) |
| 23 | \( 1 + 0.701T + 23T^{2} \) |
| 29 | \( 1 - 5.04T + 29T^{2} \) |
| 31 | \( 1 + 7.26T + 31T^{2} \) |
| 37 | \( 1 - 2.08T + 37T^{2} \) |
| 41 | \( 1 + 1.03T + 41T^{2} \) |
| 43 | \( 1 + 5.48T + 43T^{2} \) |
| 47 | \( 1 + 3.75T + 47T^{2} \) |
| 53 | \( 1 + 0.502T + 53T^{2} \) |
| 59 | \( 1 - 2.65T + 59T^{2} \) |
| 61 | \( 1 + 5.38T + 61T^{2} \) |
| 67 | \( 1 - 8.47T + 67T^{2} \) |
| 71 | \( 1 - 2.31T + 71T^{2} \) |
| 73 | \( 1 + 2.21T + 73T^{2} \) |
| 79 | \( 1 + 5.54T + 79T^{2} \) |
| 83 | \( 1 + 14.5T + 83T^{2} \) |
| 89 | \( 1 - 1.80T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
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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
−8.024141478794055516398947909333, −6.96484247772884881127287645414, −6.21192791195489747490143496887, −5.60633526958830095981297042052, −4.87498941031878733439675333698, −4.02290741327592396094541255163, −3.31318899305983818402999414100, −2.52766947556852410395817094310, −1.49070717794361119798046986645, 0,
1.49070717794361119798046986645, 2.52766947556852410395817094310, 3.31318899305983818402999414100, 4.02290741327592396094541255163, 4.87498941031878733439675333698, 5.60633526958830095981297042052, 6.21192791195489747490143496887, 6.96484247772884881127287645414, 8.024141478794055516398947909333
