| L(s) = 1 | + 1.43·2-s + 0.109·3-s + 0.0689·4-s + 5-s + 0.158·6-s − 0.695·7-s − 2.77·8-s − 2.98·9-s + 1.43·10-s − 4.85·11-s + 0.00758·12-s + 5.63·13-s − 0.999·14-s + 0.109·15-s − 4.13·16-s − 4.29·18-s − 2.32·19-s + 0.0689·20-s − 0.0763·21-s − 6.97·22-s − 4.63·23-s − 0.305·24-s + 25-s + 8.11·26-s − 0.658·27-s − 0.0479·28-s − 6.50·29-s + ⋯ |
| L(s) = 1 | + 1.01·2-s + 0.0634·3-s + 0.0344·4-s + 0.447·5-s + 0.0645·6-s − 0.262·7-s − 0.982·8-s − 0.995·9-s + 0.454·10-s − 1.46·11-s + 0.00218·12-s + 1.56·13-s − 0.267·14-s + 0.0283·15-s − 1.03·16-s − 1.01·18-s − 0.532·19-s + 0.0154·20-s − 0.0166·21-s − 1.48·22-s − 0.966·23-s − 0.0623·24-s + 0.200·25-s + 1.59·26-s − 0.126·27-s − 0.00906·28-s − 1.20·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 1.43T + 2T^{2} \) |
| 3 | \( 1 - 0.109T + 3T^{2} \) |
| 7 | \( 1 + 0.695T + 7T^{2} \) |
| 11 | \( 1 + 4.85T + 11T^{2} \) |
| 13 | \( 1 - 5.63T + 13T^{2} \) |
| 19 | \( 1 + 2.32T + 19T^{2} \) |
| 23 | \( 1 + 4.63T + 23T^{2} \) |
| 29 | \( 1 + 6.50T + 29T^{2} \) |
| 31 | \( 1 + 6.63T + 31T^{2} \) |
| 37 | \( 1 + 0.118T + 37T^{2} \) |
| 41 | \( 1 + 1.07T + 41T^{2} \) |
| 43 | \( 1 + 0.641T + 43T^{2} \) |
| 47 | \( 1 - 4.93T + 47T^{2} \) |
| 53 | \( 1 - 11.9T + 53T^{2} \) |
| 59 | \( 1 + 9.91T + 59T^{2} \) |
| 61 | \( 1 + 1.60T + 61T^{2} \) |
| 67 | \( 1 + 2.99T + 67T^{2} \) |
| 71 | \( 1 - 4.68T + 71T^{2} \) |
| 73 | \( 1 + 5.49T + 73T^{2} \) |
| 79 | \( 1 + 14.8T + 79T^{2} \) |
| 83 | \( 1 - 5.03T + 83T^{2} \) |
| 89 | \( 1 + 2.35T + 89T^{2} \) |
| 97 | \( 1 - 2.70T + 97T^{2} \) |
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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
−8.927723811903568177384385571750, −8.512627723419375319202666457544, −7.47087468708918540895200453255, −6.05245743115703817890495089925, −5.89002518183519086505690580304, −5.04948137468406102007760861371, −3.89165955074959006601571915719, −3.16373498657202430105749384750, −2.14404770043725092155692082593, 0,
2.14404770043725092155692082593, 3.16373498657202430105749384750, 3.89165955074959006601571915719, 5.04948137468406102007760861371, 5.89002518183519086505690580304, 6.05245743115703817890495089925, 7.47087468708918540895200453255, 8.512627723419375319202666457544, 8.927723811903568177384385571750
