L(s) = 1 | + 2-s − 2.47·3-s + 4-s − 2.86·5-s − 2.47·6-s − 1.77·7-s + 8-s + 3.12·9-s − 2.86·10-s − 0.764·11-s − 2.47·12-s − 1.77·14-s + 7.08·15-s + 16-s + 0.468·17-s + 3.12·18-s − 19-s − 2.86·20-s + 4.38·21-s − 0.764·22-s − 3.33·23-s − 2.47·24-s + 3.20·25-s − 0.309·27-s − 1.77·28-s + 3.54·29-s + 7.08·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.42·3-s + 0.5·4-s − 1.28·5-s − 1.01·6-s − 0.669·7-s + 0.353·8-s + 1.04·9-s − 0.905·10-s − 0.230·11-s − 0.714·12-s − 0.473·14-s + 1.82·15-s + 0.250·16-s + 0.113·17-s + 0.736·18-s − 0.229·19-s − 0.640·20-s + 0.956·21-s − 0.162·22-s − 0.695·23-s − 0.505·24-s + 0.640·25-s − 0.0595·27-s − 0.334·28-s + 0.658·29-s + 1.29·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + 2.47T + 3T^{2} \) |
| 5 | \( 1 + 2.86T + 5T^{2} \) |
| 7 | \( 1 + 1.77T + 7T^{2} \) |
| 11 | \( 1 + 0.764T + 11T^{2} \) |
| 17 | \( 1 - 0.468T + 17T^{2} \) |
| 23 | \( 1 + 3.33T + 23T^{2} \) |
| 29 | \( 1 - 3.54T + 29T^{2} \) |
| 31 | \( 1 - 2.73T + 31T^{2} \) |
| 37 | \( 1 - 5.74T + 37T^{2} \) |
| 41 | \( 1 - 9.46T + 41T^{2} \) |
| 43 | \( 1 - 6.78T + 43T^{2} \) |
| 47 | \( 1 - 1.37T + 47T^{2} \) |
| 53 | \( 1 + 9.55T + 53T^{2} \) |
| 59 | \( 1 + 5.76T + 59T^{2} \) |
| 61 | \( 1 + 2.36T + 61T^{2} \) |
| 67 | \( 1 - 1.49T + 67T^{2} \) |
| 71 | \( 1 - 2.41T + 71T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 + 9.31T + 79T^{2} \) |
| 83 | \( 1 - 3.26T + 83T^{2} \) |
| 89 | \( 1 - 14.5T + 89T^{2} \) |
| 97 | \( 1 - 2.74T + 97T^{2} \) |
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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
−7.67792466556926598660374945868, −6.60218800483563663149757999444, −6.28509412454468102402089863810, −5.57133763943317386153187511275, −4.68640881284471929080978653589, −4.25253919589959095115449168818, −3.43244241884831653575005231192, −2.52633151734431656973299303293, −0.965552607878890434418454202524, 0,
0.965552607878890434418454202524, 2.52633151734431656973299303293, 3.43244241884831653575005231192, 4.25253919589959095115449168818, 4.68640881284471929080978653589, 5.57133763943317386153187511275, 6.28509412454468102402089863810, 6.60218800483563663149757999444, 7.67792466556926598660374945868