| L(s) = 1 | + 2-s − 1.26·3-s + 4-s + 2.97·5-s − 1.26·6-s + 1.38·7-s + 8-s − 1.40·9-s + 2.97·10-s − 4.03·11-s − 1.26·12-s − 5.15·13-s + 1.38·14-s − 3.75·15-s + 16-s + 4.51·17-s − 1.40·18-s − 8.09·19-s + 2.97·20-s − 1.74·21-s − 4.03·22-s + 3.34·23-s − 1.26·24-s + 3.86·25-s − 5.15·26-s + 5.56·27-s + 1.38·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.728·3-s + 0.5·4-s + 1.33·5-s − 0.515·6-s + 0.522·7-s + 0.353·8-s − 0.468·9-s + 0.941·10-s − 1.21·11-s − 0.364·12-s − 1.42·13-s + 0.369·14-s − 0.970·15-s + 0.250·16-s + 1.09·17-s − 0.331·18-s − 1.85·19-s + 0.665·20-s − 0.380·21-s − 0.859·22-s + 0.696·23-s − 0.257·24-s + 0.772·25-s − 1.01·26-s + 1.07·27-s + 0.261·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 + T \) |
| good | 3 | \( 1 + 1.26T + 3T^{2} \) |
| 5 | \( 1 - 2.97T + 5T^{2} \) |
| 7 | \( 1 - 1.38T + 7T^{2} \) |
| 11 | \( 1 + 4.03T + 11T^{2} \) |
| 13 | \( 1 + 5.15T + 13T^{2} \) |
| 17 | \( 1 - 4.51T + 17T^{2} \) |
| 19 | \( 1 + 8.09T + 19T^{2} \) |
| 23 | \( 1 - 3.34T + 23T^{2} \) |
| 29 | \( 1 + 3.63T + 29T^{2} \) |
| 37 | \( 1 - 11.0T + 37T^{2} \) |
| 41 | \( 1 - 2.12T + 41T^{2} \) |
| 43 | \( 1 + 5.77T + 43T^{2} \) |
| 47 | \( 1 + 1.60T + 47T^{2} \) |
| 53 | \( 1 + 9.85T + 53T^{2} \) |
| 59 | \( 1 + 2.44T + 59T^{2} \) |
| 61 | \( 1 - 5.95T + 61T^{2} \) |
| 67 | \( 1 + 15.0T + 67T^{2} \) |
| 71 | \( 1 + 7.81T + 71T^{2} \) |
| 73 | \( 1 - 2.73T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 + 2.48T + 83T^{2} \) |
| 89 | \( 1 + 0.210T + 89T^{2} \) |
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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
−7.67330972575973517275347264650, −6.73806362225705972092342178864, −6.03014826924530005997564425903, −5.56582810475294266399585857062, −4.97316890836946655664470419626, −4.47263309676617861925411932049, −2.93249059591969957161114910451, −2.47966387283124978499019401498, −1.56417001808961868047218337856, 0,
1.56417001808961868047218337856, 2.47966387283124978499019401498, 2.93249059591969957161114910451, 4.47263309676617861925411932049, 4.97316890836946655664470419626, 5.56582810475294266399585857062, 6.03014826924530005997564425903, 6.73806362225705972092342178864, 7.67330972575973517275347264650
