L(s) = 1 | + 1.40·2-s + 1.06·3-s − 0.0283·4-s + 0.529·5-s + 1.49·6-s − 2.26·7-s − 2.84·8-s − 1.87·9-s + 0.743·10-s − 2.50·11-s − 0.0301·12-s − 13-s − 3.17·14-s + 0.562·15-s − 3.94·16-s − 2.07·17-s − 2.62·18-s − 2.86·19-s − 0.0150·20-s − 2.40·21-s − 3.52·22-s + 7.90·23-s − 3.02·24-s − 4.71·25-s − 1.40·26-s − 5.17·27-s + 0.0642·28-s + ⋯ |
L(s) = 1 | + 0.992·2-s + 0.613·3-s − 0.0141·4-s + 0.236·5-s + 0.608·6-s − 0.854·7-s − 1.00·8-s − 0.623·9-s + 0.235·10-s − 0.756·11-s − 0.00870·12-s − 0.277·13-s − 0.848·14-s + 0.145·15-s − 0.985·16-s − 0.503·17-s − 0.619·18-s − 0.658·19-s − 0.00336·20-s − 0.524·21-s − 0.751·22-s + 1.64·23-s − 0.617·24-s − 0.943·25-s − 0.275·26-s − 0.995·27-s + 0.0121·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
good | 2 | \( 1 - 1.40T + 2T^{2} \) |
| 3 | \( 1 - 1.06T + 3T^{2} \) |
| 5 | \( 1 - 0.529T + 5T^{2} \) |
| 7 | \( 1 + 2.26T + 7T^{2} \) |
| 11 | \( 1 + 2.50T + 11T^{2} \) |
| 17 | \( 1 + 2.07T + 17T^{2} \) |
| 19 | \( 1 + 2.86T + 19T^{2} \) |
| 23 | \( 1 - 7.90T + 23T^{2} \) |
| 29 | \( 1 + 2.39T + 29T^{2} \) |
| 31 | \( 1 + 4.34T + 31T^{2} \) |
| 37 | \( 1 - 11.0T + 37T^{2} \) |
| 41 | \( 1 + 1.58T + 41T^{2} \) |
| 43 | \( 1 - 3.50T + 43T^{2} \) |
| 47 | \( 1 + 6.19T + 47T^{2} \) |
| 53 | \( 1 + 0.857T + 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 + 4.15T + 61T^{2} \) |
| 67 | \( 1 + 12.6T + 67T^{2} \) |
| 71 | \( 1 - 6.26T + 71T^{2} \) |
| 73 | \( 1 - 4.92T + 73T^{2} \) |
| 79 | \( 1 + 6.34T + 79T^{2} \) |
| 83 | \( 1 + 3.69T + 83T^{2} \) |
| 89 | \( 1 - 3.51T + 89T^{2} \) |
| 97 | \( 1 + 6.39T + 97T^{2} \) |
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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
−9.236564234647355045354556899644, −8.531556999079847711570709723354, −7.55314928573131172152853417219, −6.47984357501151182229388896669, −5.76429746857390220200467434311, −4.96172026409521192888452769397, −3.93048631915628247746754852028, −3.04472261890513224561338128113, −2.39552044994281954725691531879, 0,
2.39552044994281954725691531879, 3.04472261890513224561338128113, 3.93048631915628247746754852028, 4.96172026409521192888452769397, 5.76429746857390220200467434311, 6.47984357501151182229388896669, 7.55314928573131172152853417219, 8.531556999079847711570709723354, 9.236564234647355045354556899644