L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s − 2.20·5-s − 6·6-s − 7·7-s − 8·8-s + 9·9-s + 4.40·10-s + 29.6·11-s + 12·12-s + 23.3·13-s + 14·14-s − 6.61·15-s + 16·16-s − 96.0·17-s − 18·18-s − 56.0·19-s − 8.81·20-s − 21·21-s − 59.3·22-s − 23·23-s − 24·24-s − 120.·25-s − 46.7·26-s + 27·27-s − 28·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.197·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.139·10-s + 0.813·11-s + 0.288·12-s + 0.499·13-s + 0.267·14-s − 0.113·15-s + 0.250·16-s − 1.37·17-s − 0.235·18-s − 0.676·19-s − 0.0985·20-s − 0.218·21-s − 0.575·22-s − 0.208·23-s − 0.204·24-s − 0.961·25-s − 0.352·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + 2T \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 + 7T \) |
| 23 | \( 1 + 23T \) |
good | 5 | \( 1 + 2.20T + 125T^{2} \) |
| 11 | \( 1 - 29.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 23.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 96.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 56.0T + 6.85e3T^{2} \) |
| 29 | \( 1 - 210.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 129.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 108.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 195.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 224.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 126.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 382.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 648.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 47.6T + 2.26e5T^{2} \) |
| 67 | \( 1 + 520.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 994.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 76.6T + 3.89e5T^{2} \) |
| 79 | \( 1 + 692.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.28e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 564.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 428.T + 9.12e5T^{2} \) |
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
−8.930272509626874648180241629934, −8.680356946734554945619299539364, −7.69571068465633472118578979824, −6.67756507791107982977463596390, −6.20583204534692070214944982954, −4.57334580757735090486056625340, −3.68275665359415063708540783624, −2.54646127836588584059394727671, −1.45178833613447801449564461512, 0,
1.45178833613447801449564461512, 2.54646127836588584059394727671, 3.68275665359415063708540783624, 4.57334580757735090486056625340, 6.20583204534692070214944982954, 6.67756507791107982977463596390, 7.69571068465633472118578979824, 8.680356946734554945619299539364, 8.930272509626874648180241629934