| L(s) = 1 | + 0.154·2-s + 0.533·3-s − 1.97·4-s − 5-s + 0.0826·6-s + 2.26·7-s − 0.615·8-s − 2.71·9-s − 0.154·10-s − 3.52·11-s − 1.05·12-s + 1.90·13-s + 0.350·14-s − 0.533·15-s + 3.85·16-s + 6.24·17-s − 0.420·18-s + 3.99·19-s + 1.97·20-s + 1.20·21-s − 0.546·22-s − 2.76·23-s − 0.328·24-s + 25-s + 0.294·26-s − 3.05·27-s − 4.47·28-s + ⋯ |
| L(s) = 1 | + 0.109·2-s + 0.308·3-s − 0.988·4-s − 0.447·5-s + 0.0337·6-s + 0.855·7-s − 0.217·8-s − 0.905·9-s − 0.0489·10-s − 1.06·11-s − 0.304·12-s + 0.527·13-s + 0.0936·14-s − 0.137·15-s + 0.964·16-s + 1.51·17-s − 0.0991·18-s + 0.916·19-s + 0.441·20-s + 0.263·21-s − 0.116·22-s − 0.575·23-s − 0.0671·24-s + 0.200·25-s + 0.0578·26-s − 0.587·27-s − 0.845·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 - 0.154T + 2T^{2} \) |
| 3 | \( 1 - 0.533T + 3T^{2} \) |
| 7 | \( 1 - 2.26T + 7T^{2} \) |
| 11 | \( 1 + 3.52T + 11T^{2} \) |
| 13 | \( 1 - 1.90T + 13T^{2} \) |
| 17 | \( 1 - 6.24T + 17T^{2} \) |
| 19 | \( 1 - 3.99T + 19T^{2} \) |
| 23 | \( 1 + 2.76T + 23T^{2} \) |
| 29 | \( 1 + 8.23T + 29T^{2} \) |
| 31 | \( 1 - 8.66T + 31T^{2} \) |
| 37 | \( 1 + 7.40T + 37T^{2} \) |
| 41 | \( 1 + 5.38T + 41T^{2} \) |
| 43 | \( 1 + 9.62T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 + 5.84T + 53T^{2} \) |
| 59 | \( 1 + 9.55T + 59T^{2} \) |
| 61 | \( 1 + 8.52T + 61T^{2} \) |
| 67 | \( 1 + 6.38T + 67T^{2} \) |
| 71 | \( 1 - 5.92T + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 - 14.8T + 79T^{2} \) |
| 83 | \( 1 + 13.7T + 83T^{2} \) |
| 89 | \( 1 - 9.87T + 89T^{2} \) |
| 97 | \( 1 - 9.69T + 97T^{2} \) |
| 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
−8.547528129318372921786672141796, −7.949683032859562307860591008000, −7.77120360860417930018150341448, −6.19312421040886234643092800615, −5.19539147947983512617590200117, −4.98330247125081512505933736925, −3.57482033633998237284492524247, −3.14050175549776228681212595802, −1.52742836087953743382827794353, 0,
1.52742836087953743382827794353, 3.14050175549776228681212595802, 3.57482033633998237284492524247, 4.98330247125081512505933736925, 5.19539147947983512617590200117, 6.19312421040886234643092800615, 7.77120360860417930018150341448, 7.949683032859562307860591008000, 8.547528129318372921786672141796
