| L(s) = 1 | + 3.26·2-s − 3·3-s + 2.63·4-s + 7.83·5-s − 9.78·6-s − 27.8·7-s − 17.4·8-s + 9·9-s + 25.5·10-s − 30.6·11-s − 7.90·12-s + 8.76·13-s − 90.9·14-s − 23.5·15-s − 78.1·16-s + 40.8·17-s + 29.3·18-s − 159.·19-s + 20.6·20-s + 83.6·21-s − 99.9·22-s + 11.8·23-s + 52.4·24-s − 63.6·25-s + 28.5·26-s − 27·27-s − 73.5·28-s + ⋯ |
| L(s) = 1 | + 1.15·2-s − 0.577·3-s + 0.329·4-s + 0.700·5-s − 0.665·6-s − 1.50·7-s − 0.773·8-s + 0.333·9-s + 0.807·10-s − 0.839·11-s − 0.190·12-s + 0.187·13-s − 1.73·14-s − 0.404·15-s − 1.22·16-s + 0.582·17-s + 0.384·18-s − 1.92·19-s + 0.230·20-s + 0.869·21-s − 0.968·22-s + 0.107·23-s + 0.446·24-s − 0.509·25-s + 0.215·26-s − 0.192·27-s − 0.496·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 67 | \( 1 - 67T \) |
| good | 2 | \( 1 - 3.26T + 8T^{2} \) |
| 5 | \( 1 - 7.83T + 125T^{2} \) |
| 7 | \( 1 + 27.8T + 343T^{2} \) |
| 11 | \( 1 + 30.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 8.76T + 2.19e3T^{2} \) |
| 17 | \( 1 - 40.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 159.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 11.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 229.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 45.2T + 2.97e4T^{2} \) |
| 37 | \( 1 + 38.6T + 5.06e4T^{2} \) |
| 41 | \( 1 + 243.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 154.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 97.0T + 1.03e5T^{2} \) |
| 53 | \( 1 - 142.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 256.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 573.T + 2.26e5T^{2} \) |
| 71 | \( 1 - 522.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 913.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 513.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 687.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 385.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 109.T + 9.12e5T^{2} \) |
| 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
−11.91102498752233014421439963351, −10.47897077492901832323148242959, −9.832817351302248183327841617550, −8.587483199186097013546793870676, −6.71109322714022720643948077395, −6.10066470819373576559670832206, −5.18122565366868498781456821423, −3.86457209866958767910738699770, −2.59190208003117246628181082690, 0,
2.59190208003117246628181082690, 3.86457209866958767910738699770, 5.18122565366868498781456821423, 6.10066470819373576559670832206, 6.71109322714022720643948077395, 8.587483199186097013546793870676, 9.832817351302248183327841617550, 10.47897077492901832323148242959, 11.91102498752233014421439963351
