L(s) = 1 | + 2.10·2-s + 3-s + 2.41·4-s − 0.968·5-s + 2.10·6-s − 2.79·7-s + 0.874·8-s + 9-s − 2.03·10-s + 1.65·11-s + 2.41·12-s − 3.84·13-s − 5.87·14-s − 0.968·15-s − 2.99·16-s + 17-s + 2.10·18-s − 6.42·19-s − 2.33·20-s − 2.79·21-s + 3.47·22-s + 5.76·23-s + 0.874·24-s − 4.06·25-s − 8.08·26-s + 27-s − 6.75·28-s + ⋯ |
L(s) = 1 | + 1.48·2-s + 0.577·3-s + 1.20·4-s − 0.432·5-s + 0.857·6-s − 1.05·7-s + 0.309·8-s + 0.333·9-s − 0.643·10-s + 0.498·11-s + 0.697·12-s − 1.06·13-s − 1.56·14-s − 0.249·15-s − 0.748·16-s + 0.242·17-s + 0.495·18-s − 1.47·19-s − 0.523·20-s − 0.609·21-s + 0.740·22-s + 1.20·23-s + 0.178·24-s − 0.812·25-s − 1.58·26-s + 0.192·27-s − 1.27·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 79 | \( 1 + T \) |
good | 2 | \( 1 - 2.10T + 2T^{2} \) |
| 5 | \( 1 + 0.968T + 5T^{2} \) |
| 7 | \( 1 + 2.79T + 7T^{2} \) |
| 11 | \( 1 - 1.65T + 11T^{2} \) |
| 13 | \( 1 + 3.84T + 13T^{2} \) |
| 19 | \( 1 + 6.42T + 19T^{2} \) |
| 23 | \( 1 - 5.76T + 23T^{2} \) |
| 29 | \( 1 + 1.39T + 29T^{2} \) |
| 31 | \( 1 - 5.58T + 31T^{2} \) |
| 37 | \( 1 + 4.64T + 37T^{2} \) |
| 41 | \( 1 + 4.80T + 41T^{2} \) |
| 43 | \( 1 + 1.63T + 43T^{2} \) |
| 47 | \( 1 + 3.46T + 47T^{2} \) |
| 53 | \( 1 + 6.66T + 53T^{2} \) |
| 59 | \( 1 - 7.59T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 + 11.0T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 + 16.3T + 73T^{2} \) |
| 83 | \( 1 - 3.58T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 - 1.40T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.947547797213313807991247898633, −6.96971340406315118679290418382, −6.61964258842876210021689618764, −5.80047855126359073623897690988, −4.79588277141546883702913704144, −4.27641958162716356684584282296, −3.39749001865564538371758414071, −2.94486086206691247830369203434, −1.94655287168196823238305832513, 0,
1.94655287168196823238305832513, 2.94486086206691247830369203434, 3.39749001865564538371758414071, 4.27641958162716356684584282296, 4.79588277141546883702913704144, 5.80047855126359073623897690988, 6.61964258842876210021689618764, 6.96971340406315118679290418382, 7.947547797213313807991247898633