| L(s) = 1 | + 0.125·2-s − 3.53·3-s − 7.98·4-s + 5·5-s − 0.442·6-s − 25.7·7-s − 2.00·8-s − 14.5·9-s + 0.626·10-s + 11·11-s + 28.2·12-s − 9.72·13-s − 3.22·14-s − 17.6·15-s + 63.6·16-s + 102.·17-s − 1.82·18-s + 19·19-s − 39.9·20-s + 90.8·21-s + 1.37·22-s + 119.·23-s + 7.07·24-s + 25·25-s − 1.21·26-s + 146.·27-s + 205.·28-s + ⋯ |
| L(s) = 1 | + 0.0443·2-s − 0.679·3-s − 0.998·4-s + 0.447·5-s − 0.0301·6-s − 1.38·7-s − 0.0885·8-s − 0.537·9-s + 0.0198·10-s + 0.301·11-s + 0.678·12-s − 0.207·13-s − 0.0615·14-s − 0.304·15-s + 0.994·16-s + 1.46·17-s − 0.0238·18-s + 0.229·19-s − 0.446·20-s + 0.943·21-s + 0.0133·22-s + 1.08·23-s + 0.0601·24-s + 0.200·25-s − 0.00919·26-s + 1.04·27-s + 1.38·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - 5T \) |
| 11 | \( 1 - 11T \) |
| 19 | \( 1 - 19T \) |
| good | 2 | \( 1 - 0.125T + 8T^{2} \) |
| 3 | \( 1 + 3.53T + 27T^{2} \) |
| 7 | \( 1 + 25.7T + 343T^{2} \) |
| 13 | \( 1 + 9.72T + 2.19e3T^{2} \) |
| 17 | \( 1 - 102.T + 4.91e3T^{2} \) |
| 23 | \( 1 - 119.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 270.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 68.5T + 2.97e4T^{2} \) |
| 37 | \( 1 - 383.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 97.2T + 6.89e4T^{2} \) |
| 43 | \( 1 + 516.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 102.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 136.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 106.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 359.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 843.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 442.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 675.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 547.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 339.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.47e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 593.T + 9.12e5T^{2} \) |
| 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
−9.414012029118255626495301855406, −8.470451805186986798080520718782, −7.36398800933480071361873942639, −6.31358758471193204241036302225, −5.67027667463681110190185366281, −4.99579002946050157072592842430, −3.67465612081772467875837726041, −2.93900508026705185912208101847, −1.02723008815609245747743557830, 0,
1.02723008815609245747743557830, 2.93900508026705185912208101847, 3.67465612081772467875837726041, 4.99579002946050157072592842430, 5.67027667463681110190185366281, 6.31358758471193204241036302225, 7.36398800933480071361873942639, 8.470451805186986798080520718782, 9.414012029118255626495301855406
