L(s) = 1 | − 2.61·2-s + 3-s + 4.85·4-s + 0.391·5-s − 2.61·6-s + 2.70·7-s − 7.48·8-s + 9-s − 1.02·10-s − 3.87·11-s + 4.85·12-s − 0.879·13-s − 7.08·14-s + 0.391·15-s + 9.87·16-s + 17-s − 2.61·18-s + 0.781·19-s + 1.90·20-s + 2.70·21-s + 10.1·22-s − 7.15·23-s − 7.48·24-s − 4.84·25-s + 2.30·26-s + 27-s + 13.1·28-s + ⋯ |
L(s) = 1 | − 1.85·2-s + 0.577·3-s + 2.42·4-s + 0.175·5-s − 1.06·6-s + 1.02·7-s − 2.64·8-s + 0.333·9-s − 0.324·10-s − 1.16·11-s + 1.40·12-s − 0.243·13-s − 1.89·14-s + 0.101·15-s + 2.46·16-s + 0.242·17-s − 0.617·18-s + 0.179·19-s + 0.425·20-s + 0.590·21-s + 2.16·22-s − 1.49·23-s − 1.52·24-s − 0.969·25-s + 0.451·26-s + 0.192·27-s + 2.48·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 157 | \( 1 + T \) |
good | 2 | \( 1 + 2.61T + 2T^{2} \) |
| 5 | \( 1 - 0.391T + 5T^{2} \) |
| 7 | \( 1 - 2.70T + 7T^{2} \) |
| 11 | \( 1 + 3.87T + 11T^{2} \) |
| 13 | \( 1 + 0.879T + 13T^{2} \) |
| 19 | \( 1 - 0.781T + 19T^{2} \) |
| 23 | \( 1 + 7.15T + 23T^{2} \) |
| 29 | \( 1 + 3.67T + 29T^{2} \) |
| 31 | \( 1 - 9.68T + 31T^{2} \) |
| 37 | \( 1 + 2.30T + 37T^{2} \) |
| 41 | \( 1 + 4.05T + 41T^{2} \) |
| 43 | \( 1 - 4.55T + 43T^{2} \) |
| 47 | \( 1 + 3.63T + 47T^{2} \) |
| 53 | \( 1 - 4.75T + 53T^{2} \) |
| 59 | \( 1 - 5.18T + 59T^{2} \) |
| 61 | \( 1 - 9.15T + 61T^{2} \) |
| 67 | \( 1 + 12.5T + 67T^{2} \) |
| 71 | \( 1 - 6.72T + 71T^{2} \) |
| 73 | \( 1 + 3.80T + 73T^{2} \) |
| 79 | \( 1 + 0.140T + 79T^{2} \) |
| 83 | \( 1 - 8.27T + 83T^{2} \) |
| 89 | \( 1 - 1.59T + 89T^{2} \) |
| 97 | \( 1 - 9.61T + 97T^{2} \) |
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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
−7.72859951315497781419268506127, −7.33949262359994393219405471725, −6.35854119278236801760443040425, −5.62347751433702814681479579939, −4.73805098384401671387295317433, −3.61827882294061452425395407631, −2.49668463712802442440885471227, −2.11732291828897970423373347536, −1.22840046759251047310472545965, 0,
1.22840046759251047310472545965, 2.11732291828897970423373347536, 2.49668463712802442440885471227, 3.61827882294061452425395407631, 4.73805098384401671387295317433, 5.62347751433702814681479579939, 6.35854119278236801760443040425, 7.33949262359994393219405471725, 7.72859951315497781419268506127