| L(s) = 1 | + 0.290·2-s + 3-s − 1.91·4-s + 5-s + 0.290·6-s + 1.24·7-s − 1.13·8-s + 9-s + 0.290·10-s − 3.12·11-s − 1.91·12-s + 13-s + 0.361·14-s + 15-s + 3.49·16-s + 0.845·17-s + 0.290·18-s + 0.330·19-s − 1.91·20-s + 1.24·21-s − 0.908·22-s − 3.72·23-s − 1.13·24-s + 25-s + 0.290·26-s + 27-s − 2.37·28-s + ⋯ |
| L(s) = 1 | + 0.205·2-s + 0.577·3-s − 0.957·4-s + 0.447·5-s + 0.118·6-s + 0.469·7-s − 0.402·8-s + 0.333·9-s + 0.0919·10-s − 0.942·11-s − 0.552·12-s + 0.277·13-s + 0.0965·14-s + 0.258·15-s + 0.874·16-s + 0.205·17-s + 0.0685·18-s + 0.0759·19-s − 0.428·20-s + 0.271·21-s − 0.193·22-s − 0.776·23-s − 0.232·24-s + 0.200·25-s + 0.0570·26-s + 0.192·27-s − 0.449·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - 0.290T + 2T^{2} \) |
| 7 | \( 1 - 1.24T + 7T^{2} \) |
| 11 | \( 1 + 3.12T + 11T^{2} \) |
| 17 | \( 1 - 0.845T + 17T^{2} \) |
| 19 | \( 1 - 0.330T + 19T^{2} \) |
| 23 | \( 1 + 3.72T + 23T^{2} \) |
| 29 | \( 1 + 10.1T + 29T^{2} \) |
| 37 | \( 1 + 1.38T + 37T^{2} \) |
| 41 | \( 1 + 5.15T + 41T^{2} \) |
| 43 | \( 1 - 1.00T + 43T^{2} \) |
| 47 | \( 1 + 8.00T + 47T^{2} \) |
| 53 | \( 1 + 7.87T + 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 - 10.2T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 + 8.14T + 71T^{2} \) |
| 73 | \( 1 + 0.607T + 73T^{2} \) |
| 79 | \( 1 + 0.654T + 79T^{2} \) |
| 83 | \( 1 - 6.92T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 + 2.62T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.018829395164198486069456425622, −7.14829171025589086197838208771, −6.14106820247769802329703602439, −5.36147678805365496901917273479, −4.94567858368552943914789400213, −3.96028928572102282135795450897, −3.37222588117356697038528720362, −2.34521350937610748393618828898, −1.44829725828415336920708363136, 0,
1.44829725828415336920708363136, 2.34521350937610748393618828898, 3.37222588117356697038528720362, 3.96028928572102282135795450897, 4.94567858368552943914789400213, 5.36147678805365496901917273479, 6.14106820247769802329703602439, 7.14829171025589086197838208771, 8.018829395164198486069456425622
