L(s) = 1 | + 2-s − 2.06·3-s + 4-s + 5-s − 2.06·6-s + 0.701·7-s + 8-s + 1.27·9-s + 10-s − 4.56·11-s − 2.06·12-s − 13-s + 0.701·14-s − 2.06·15-s + 16-s + 7.12·17-s + 1.27·18-s − 6.61·19-s + 20-s − 1.45·21-s − 4.56·22-s + 3.53·23-s − 2.06·24-s + 25-s − 26-s + 3.55·27-s + 0.701·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.19·3-s + 0.5·4-s + 0.447·5-s − 0.844·6-s + 0.265·7-s + 0.353·8-s + 0.426·9-s + 0.316·10-s − 1.37·11-s − 0.597·12-s − 0.277·13-s + 0.187·14-s − 0.534·15-s + 0.250·16-s + 1.72·17-s + 0.301·18-s − 1.51·19-s + 0.223·20-s − 0.316·21-s − 0.974·22-s + 0.737·23-s − 0.422·24-s + 0.200·25-s − 0.196·26-s + 0.685·27-s + 0.132·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 + T \) |
good | 3 | \( 1 + 2.06T + 3T^{2} \) |
| 7 | \( 1 - 0.701T + 7T^{2} \) |
| 11 | \( 1 + 4.56T + 11T^{2} \) |
| 17 | \( 1 - 7.12T + 17T^{2} \) |
| 19 | \( 1 + 6.61T + 19T^{2} \) |
| 23 | \( 1 - 3.53T + 23T^{2} \) |
| 29 | \( 1 + 1.57T + 29T^{2} \) |
| 37 | \( 1 + 1.23T + 37T^{2} \) |
| 41 | \( 1 - 3.18T + 41T^{2} \) |
| 43 | \( 1 + 5.91T + 43T^{2} \) |
| 47 | \( 1 + 5.32T + 47T^{2} \) |
| 53 | \( 1 + 3.23T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 - 2.53T + 61T^{2} \) |
| 67 | \( 1 - 5.14T + 67T^{2} \) |
| 71 | \( 1 + 3.62T + 71T^{2} \) |
| 73 | \( 1 + 14.7T + 73T^{2} \) |
| 79 | \( 1 + 1.62T + 79T^{2} \) |
| 83 | \( 1 + 6.10T + 83T^{2} \) |
| 89 | \( 1 - 13.7T + 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
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
−7.88327007713128977658157475422, −7.19297401196295751549720939465, −6.24763426462944770271954027705, −5.81383049798487493026225372432, −5.02131268623076049431122666088, −4.76335819656839770594136562883, −3.41395779453875954296860723131, −2.56666051846758708496482413492, −1.44096026793157959599940722278, 0,
1.44096026793157959599940722278, 2.56666051846758708496482413492, 3.41395779453875954296860723131, 4.76335819656839770594136562883, 5.02131268623076049431122666088, 5.81383049798487493026225372432, 6.24763426462944770271954027705, 7.19297401196295751549720939465, 7.88327007713128977658157475422