| L(s) = 1 | − 2.56i·2-s − 4.56·4-s − 0.561i·5-s + 3.56i·7-s + 6.56i·8-s − 1.43·10-s − 2i·11-s + 9.12·14-s + 7.68·16-s + 2.56·17-s − 1.12i·19-s + 2.56i·20-s − 5.12·22-s + 2·23-s + 4.68·25-s + ⋯ |
| L(s) = 1 | − 1.81i·2-s − 2.28·4-s − 0.251i·5-s + 1.34i·7-s + 2.31i·8-s − 0.454·10-s − 0.603i·11-s + 2.43·14-s + 1.92·16-s + 0.621·17-s − 0.257i·19-s + 0.572i·20-s − 1.09·22-s + 0.417·23-s + 0.936·25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.393000473\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.393000473\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.56iT - 2T^{2} \) |
| 5 | \( 1 + 0.561iT - 5T^{2} \) |
| 7 | \( 1 - 3.56iT - 7T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 17 | \( 1 - 2.56T + 17T^{2} \) |
| 19 | \( 1 + 1.12iT - 19T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 29 | \( 1 - 5.68T + 29T^{2} \) |
| 31 | \( 1 + 1.56iT - 31T^{2} \) |
| 37 | \( 1 + 3.43iT - 37T^{2} \) |
| 41 | \( 1 + 2.56iT - 41T^{2} \) |
| 43 | \( 1 + 0.438T + 43T^{2} \) |
| 47 | \( 1 + 8.24iT - 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 + 11.1iT - 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 - 0.438iT - 67T^{2} \) |
| 71 | \( 1 + 14iT - 71T^{2} \) |
| 73 | \( 1 - 1.87iT - 73T^{2} \) |
| 79 | \( 1 - 9.56T + 79T^{2} \) |
| 83 | \( 1 - 9.12iT - 83T^{2} \) |
| 89 | \( 1 - 13.1iT - 89T^{2} \) |
| 97 | \( 1 + 4.43iT - 97T^{2} \) |
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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
−9.266129835345442565130994630556, −8.715947185783872867723942065395, −8.079665779296461446817990337150, −6.53188151316221986926612149109, −5.39244292840511893873551545974, −4.87111174822761321620364046867, −3.62048825687351226800046125626, −2.86683408246551183046732003024, −2.02155372941901206529855282017, −0.73891486913757544781750861954,
1.01863710236056831658686766935, 3.20798880525452539633030946132, 4.34974275116811973382139400763, 4.84936111500598726048211830937, 5.94040065779220960938064780560, 6.79443041912126929474440776399, 7.21148761450085988000487295132, 7.934826279579463238017875635665, 8.650748181837649520796043063059, 9.685448383397413733161735836382
