| L(s) = 1 | + 4.49·2-s + 12.1·4-s − 8.19·5-s + 18.8·8-s − 36.8·10-s − 33.3·11-s + 54.5·13-s − 13.0·16-s − 1.19·17-s + 124.·19-s − 99.8·20-s − 150.·22-s − 133.·23-s − 57.7·25-s + 245.·26-s − 145.·29-s − 78.1·31-s − 208.·32-s − 5.38·34-s − 134.·37-s + 558.·38-s − 154.·40-s − 178.·41-s + 211.·43-s − 406.·44-s − 602.·46-s − 518.·47-s + ⋯ |
| L(s) = 1 | + 1.58·2-s + 1.52·4-s − 0.733·5-s + 0.830·8-s − 1.16·10-s − 0.915·11-s + 1.16·13-s − 0.203·16-s − 0.0171·17-s + 1.50·19-s − 1.11·20-s − 1.45·22-s − 1.21·23-s − 0.462·25-s + 1.85·26-s − 0.932·29-s − 0.452·31-s − 1.15·32-s − 0.0271·34-s − 0.598·37-s + 2.38·38-s − 0.609·40-s − 0.679·41-s + 0.748·43-s − 1.39·44-s − 1.92·46-s − 1.60·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 4.49T + 8T^{2} \) |
| 5 | \( 1 + 8.19T + 125T^{2} \) |
| 11 | \( 1 + 33.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 54.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 1.19T + 4.91e3T^{2} \) |
| 19 | \( 1 - 124.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 133.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 145.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 78.1T + 2.97e4T^{2} \) |
| 37 | \( 1 + 134.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 178.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 211.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 518.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 269.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 306.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 38.0T + 2.26e5T^{2} \) |
| 67 | \( 1 - 610.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.08e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.08e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.19e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 150.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 221.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.60e3T + 9.12e5T^{2} \) |
| 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
−8.680305850150465548539146802624, −7.77505762193283109661809868763, −7.12339391432473026597989086552, −5.94942413269228364629466529952, −5.50249393359753868369871466895, −4.48319752255775413531902372464, −3.66698260874981768601655991106, −3.07726222332152190730412189863, −1.75116663222038568429291041664, 0,
1.75116663222038568429291041664, 3.07726222332152190730412189863, 3.66698260874981768601655991106, 4.48319752255775413531902372464, 5.50249393359753868369871466895, 5.94942413269228364629466529952, 7.12339391432473026597989086552, 7.77505762193283109661809868763, 8.680305850150465548539146802624
