L(s) = 1 | − 2.25·2-s + 3.06·4-s − 2.01·5-s − 1.18·7-s − 2.40·8-s + 4.52·10-s − 11-s + 5.42·13-s + 2.67·14-s − 0.728·16-s − 1.37·17-s + 6.28·19-s − 6.16·20-s + 2.25·22-s + 0.813·23-s − 0.953·25-s − 12.2·26-s − 3.64·28-s + 1.06·29-s − 1.34·31-s + 6.44·32-s + 3.10·34-s + 2.39·35-s − 3.05·37-s − 14.1·38-s + 4.83·40-s − 4.31·41-s + ⋯ |
L(s) = 1 | − 1.59·2-s + 1.53·4-s − 0.899·5-s − 0.449·7-s − 0.848·8-s + 1.43·10-s − 0.301·11-s + 1.50·13-s + 0.715·14-s − 0.182·16-s − 0.334·17-s + 1.44·19-s − 1.37·20-s + 0.479·22-s + 0.169·23-s − 0.190·25-s − 2.39·26-s − 0.689·28-s + 0.198·29-s − 0.242·31-s + 1.13·32-s + 0.531·34-s + 0.404·35-s − 0.501·37-s − 2.29·38-s + 0.763·40-s − 0.674·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8019 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8019 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 + 2.25T + 2T^{2} \) |
| 5 | \( 1 + 2.01T + 5T^{2} \) |
| 7 | \( 1 + 1.18T + 7T^{2} \) |
| 13 | \( 1 - 5.42T + 13T^{2} \) |
| 17 | \( 1 + 1.37T + 17T^{2} \) |
| 19 | \( 1 - 6.28T + 19T^{2} \) |
| 23 | \( 1 - 0.813T + 23T^{2} \) |
| 29 | \( 1 - 1.06T + 29T^{2} \) |
| 31 | \( 1 + 1.34T + 31T^{2} \) |
| 37 | \( 1 + 3.05T + 37T^{2} \) |
| 41 | \( 1 + 4.31T + 41T^{2} \) |
| 43 | \( 1 - 2.61T + 43T^{2} \) |
| 47 | \( 1 + 7.25T + 47T^{2} \) |
| 53 | \( 1 + 13.8T + 53T^{2} \) |
| 59 | \( 1 - 10.5T + 59T^{2} \) |
| 61 | \( 1 + 2.55T + 61T^{2} \) |
| 67 | \( 1 - 6.63T + 67T^{2} \) |
| 71 | \( 1 + 11.0T + 71T^{2} \) |
| 73 | \( 1 - 16.1T + 73T^{2} \) |
| 79 | \( 1 + 3.90T + 79T^{2} \) |
| 83 | \( 1 + 5.31T + 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 + 3.67T + 97T^{2} \) |
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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
−7.85041258565882671386876111106, −6.96306077417596036800329029114, −6.51204638335457360987446472968, −5.58853587581005441483383279358, −4.62155828971252087182632113844, −3.60163927154309092854892890862, −3.09821112126200308615297466586, −1.84250879779647937112421387682, −0.979206824061789961110145706029, 0,
0.979206824061789961110145706029, 1.84250879779647937112421387682, 3.09821112126200308615297466586, 3.60163927154309092854892890862, 4.62155828971252087182632113844, 5.58853587581005441483383279358, 6.51204638335457360987446472968, 6.96306077417596036800329029114, 7.85041258565882671386876111106