| L(s) = 1 | − 0.506·2-s + 3-s − 1.74·4-s − 0.0764·5-s − 0.506·6-s + 7-s + 1.89·8-s + 9-s + 0.0386·10-s + 2.12·11-s − 1.74·12-s + 3.26·13-s − 0.506·14-s − 0.0764·15-s + 2.52·16-s + 0.182·17-s − 0.506·18-s − 0.977·19-s + 0.133·20-s + 21-s − 1.07·22-s − 4.30·23-s + 1.89·24-s − 4.99·25-s − 1.65·26-s + 27-s − 1.74·28-s + ⋯ |
| L(s) = 1 | − 0.357·2-s + 0.577·3-s − 0.871·4-s − 0.0341·5-s − 0.206·6-s + 0.377·7-s + 0.669·8-s + 0.333·9-s + 0.0122·10-s + 0.641·11-s − 0.503·12-s + 0.905·13-s − 0.135·14-s − 0.0197·15-s + 0.632·16-s + 0.0441·17-s − 0.119·18-s − 0.224·19-s + 0.0298·20-s + 0.218·21-s − 0.229·22-s − 0.897·23-s + 0.386·24-s − 0.998·25-s − 0.324·26-s + 0.192·27-s − 0.329·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 + 0.506T + 2T^{2} \) |
| 5 | \( 1 + 0.0764T + 5T^{2} \) |
| 11 | \( 1 - 2.12T + 11T^{2} \) |
| 13 | \( 1 - 3.26T + 13T^{2} \) |
| 17 | \( 1 - 0.182T + 17T^{2} \) |
| 19 | \( 1 + 0.977T + 19T^{2} \) |
| 23 | \( 1 + 4.30T + 23T^{2} \) |
| 29 | \( 1 + 5.74T + 29T^{2} \) |
| 31 | \( 1 + 7.92T + 31T^{2} \) |
| 37 | \( 1 - 4.63T + 37T^{2} \) |
| 41 | \( 1 + 3.13T + 41T^{2} \) |
| 43 | \( 1 - 1.11T + 43T^{2} \) |
| 47 | \( 1 - 1.28T + 47T^{2} \) |
| 53 | \( 1 + 2.79T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 + 0.0886T + 61T^{2} \) |
| 67 | \( 1 + 5.90T + 67T^{2} \) |
| 71 | \( 1 - 1.50T + 71T^{2} \) |
| 73 | \( 1 - 7.73T + 73T^{2} \) |
| 79 | \( 1 + 13.1T + 79T^{2} \) |
| 83 | \( 1 - 1.23T + 83T^{2} \) |
| 89 | \( 1 + 6.43T + 89T^{2} \) |
| 97 | \( 1 + 0.769T + 97T^{2} \) |
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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
−7.78108894796843423384107058134, −6.99996315500898480501757537950, −5.98894273384278631578336841719, −5.47528392538487864714573756794, −4.36109993472145921534172317795, −3.98725786207089193189527007406, −3.29953841494734629288801041725, −1.95673695115793996559003280350, −1.35309474272627106005514375486, 0,
1.35309474272627106005514375486, 1.95673695115793996559003280350, 3.29953841494734629288801041725, 3.98725786207089193189527007406, 4.36109993472145921534172317795, 5.47528392538487864714573756794, 5.98894273384278631578336841719, 6.99996315500898480501757537950, 7.78108894796843423384107058134
