| L(s) = 1 | + 1.32·2-s + 0.446·3-s − 0.244·4-s + 0.929·5-s + 0.592·6-s + 4.32·7-s − 2.97·8-s − 2.80·9-s + 1.23·10-s + 2.75·11-s − 0.109·12-s − 4.09·13-s + 5.72·14-s + 0.415·15-s − 3.45·16-s + 1.02·17-s − 3.71·18-s − 2.41·19-s − 0.226·20-s + 1.93·21-s + 3.64·22-s − 1.82·23-s − 1.32·24-s − 4.13·25-s − 5.42·26-s − 2.59·27-s − 1.05·28-s + ⋯ |
| L(s) = 1 | + 0.936·2-s + 0.258·3-s − 0.122·4-s + 0.415·5-s + 0.241·6-s + 1.63·7-s − 1.05·8-s − 0.933·9-s + 0.389·10-s + 0.829·11-s − 0.0314·12-s − 1.13·13-s + 1.52·14-s + 0.107·15-s − 0.863·16-s + 0.248·17-s − 0.874·18-s − 0.553·19-s − 0.0507·20-s + 0.421·21-s + 0.777·22-s − 0.380·23-s − 0.271·24-s − 0.827·25-s − 1.06·26-s − 0.498·27-s − 0.199·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.831942312\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.831942312\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 163 | \( 1 - T \) |
| good | 2 | \( 1 - 1.32T + 2T^{2} \) |
| 3 | \( 1 - 0.446T + 3T^{2} \) |
| 5 | \( 1 - 0.929T + 5T^{2} \) |
| 7 | \( 1 - 4.32T + 7T^{2} \) |
| 11 | \( 1 - 2.75T + 11T^{2} \) |
| 13 | \( 1 + 4.09T + 13T^{2} \) |
| 17 | \( 1 - 1.02T + 17T^{2} \) |
| 19 | \( 1 + 2.41T + 19T^{2} \) |
| 23 | \( 1 + 1.82T + 23T^{2} \) |
| 29 | \( 1 + 3.14T + 29T^{2} \) |
| 31 | \( 1 + 3.03T + 31T^{2} \) |
| 37 | \( 1 - 9.58T + 37T^{2} \) |
| 41 | \( 1 - 7.10T + 41T^{2} \) |
| 43 | \( 1 - 0.819T + 43T^{2} \) |
| 47 | \( 1 + 11.2T + 47T^{2} \) |
| 53 | \( 1 - 5.83T + 53T^{2} \) |
| 59 | \( 1 - 12.5T + 59T^{2} \) |
| 61 | \( 1 - 3.42T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 - 9.38T + 71T^{2} \) |
| 73 | \( 1 + 0.472T + 73T^{2} \) |
| 79 | \( 1 + 13.1T + 79T^{2} \) |
| 83 | \( 1 - 5.38T + 83T^{2} \) |
| 89 | \( 1 - 13.6T + 89T^{2} \) |
| 97 | \( 1 + 13.5T + 97T^{2} \) |
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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
−13.01576137339878509848733971338, −11.84970244044521645075664735406, −11.32190738107564644557351909034, −9.705223796773738523568785771897, −8.732811535927424931014011558222, −7.72092109580494674023567605676, −6.01711916191861644758234756686, −5.09903510074009460337799975921, −4.03617004132020185873889644367, −2.29480665808985747365433352025,
2.29480665808985747365433352025, 4.03617004132020185873889644367, 5.09903510074009460337799975921, 6.01711916191861644758234756686, 7.72092109580494674023567605676, 8.732811535927424931014011558222, 9.705223796773738523568785771897, 11.32190738107564644557351909034, 11.84970244044521645075664735406, 13.01576137339878509848733971338
