| L(s) = 1 | − 2.74·3-s − 0.560·5-s + 1.03·7-s + 4.52·9-s + 5.17·11-s − 4.63·13-s + 1.53·15-s + 17-s − 7.68·19-s − 2.82·21-s + 1.46·23-s − 4.68·25-s − 4.17·27-s + 7.77·29-s + 2.95·31-s − 14.1·33-s − 0.577·35-s − 1.08·37-s + 12.7·39-s − 2.08·41-s − 3.40·43-s − 2.53·45-s − 9.02·47-s − 5.93·49-s − 2.74·51-s + 13.6·53-s − 2.89·55-s + ⋯ |
| L(s) = 1 | − 1.58·3-s − 0.250·5-s + 0.389·7-s + 1.50·9-s + 1.55·11-s − 1.28·13-s + 0.396·15-s + 0.242·17-s − 1.76·19-s − 0.616·21-s + 0.305·23-s − 0.937·25-s − 0.804·27-s + 1.44·29-s + 0.529·31-s − 2.47·33-s − 0.0975·35-s − 0.179·37-s + 2.03·39-s − 0.326·41-s − 0.518·43-s − 0.377·45-s − 1.31·47-s − 0.848·49-s − 0.384·51-s + 1.86·53-s − 0.390·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 + 2.74T + 3T^{2} \) |
| 5 | \( 1 + 0.560T + 5T^{2} \) |
| 7 | \( 1 - 1.03T + 7T^{2} \) |
| 11 | \( 1 - 5.17T + 11T^{2} \) |
| 13 | \( 1 + 4.63T + 13T^{2} \) |
| 19 | \( 1 + 7.68T + 19T^{2} \) |
| 23 | \( 1 - 1.46T + 23T^{2} \) |
| 29 | \( 1 - 7.77T + 29T^{2} \) |
| 31 | \( 1 - 2.95T + 31T^{2} \) |
| 37 | \( 1 + 1.08T + 37T^{2} \) |
| 41 | \( 1 + 2.08T + 41T^{2} \) |
| 43 | \( 1 + 3.40T + 43T^{2} \) |
| 47 | \( 1 + 9.02T + 47T^{2} \) |
| 53 | \( 1 - 13.6T + 53T^{2} \) |
| 61 | \( 1 - 1.43T + 61T^{2} \) |
| 67 | \( 1 - 13.5T + 67T^{2} \) |
| 71 | \( 1 + 0.0700T + 71T^{2} \) |
| 73 | \( 1 - 4.70T + 73T^{2} \) |
| 79 | \( 1 + 16.5T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 + 3.66T + 89T^{2} \) |
| 97 | \( 1 - 12.1T + 97T^{2} \) |
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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
−8.066914162414986796729162600917, −6.96278345621175243200579145573, −6.62169943418709163963462254381, −5.96048990672372561477531743606, −4.95591934832734444477839516015, −4.55422050821037637389861758433, −3.72488538958046354260281893200, −2.24150901818968277694386536918, −1.15780524866619523343240294933, 0,
1.15780524866619523343240294933, 2.24150901818968277694386536918, 3.72488538958046354260281893200, 4.55422050821037637389861758433, 4.95591934832734444477839516015, 5.96048990672372561477531743606, 6.62169943418709163963462254381, 6.96278345621175243200579145573, 8.066914162414986796729162600917
