| L(s) = 1 | − 1.29·2-s − 3-s − 0.335·4-s + 1.27·5-s + 1.29·6-s + 0.146·7-s + 3.01·8-s + 9-s − 1.64·10-s + 3.13·11-s + 0.335·12-s + 13-s − 0.189·14-s − 1.27·15-s − 3.21·16-s + 1.54·17-s − 1.29·18-s + 1.12·19-s − 0.428·20-s − 0.146·21-s − 4.04·22-s − 3.98·23-s − 3.01·24-s − 3.36·25-s − 1.29·26-s − 27-s − 0.0491·28-s + ⋯ |
| L(s) = 1 | − 0.912·2-s − 0.577·3-s − 0.167·4-s + 0.571·5-s + 0.526·6-s + 0.0554·7-s + 1.06·8-s + 0.333·9-s − 0.521·10-s + 0.945·11-s + 0.0968·12-s + 0.277·13-s − 0.0505·14-s − 0.330·15-s − 0.804·16-s + 0.375·17-s − 0.304·18-s + 0.257·19-s − 0.0958·20-s − 0.0320·21-s − 0.862·22-s − 0.830·23-s − 0.615·24-s − 0.673·25-s − 0.253·26-s − 0.192·27-s − 0.00929·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 + 1.29T + 2T^{2} \) |
| 5 | \( 1 - 1.27T + 5T^{2} \) |
| 7 | \( 1 - 0.146T + 7T^{2} \) |
| 11 | \( 1 - 3.13T + 11T^{2} \) |
| 17 | \( 1 - 1.54T + 17T^{2} \) |
| 19 | \( 1 - 1.12T + 19T^{2} \) |
| 23 | \( 1 + 3.98T + 23T^{2} \) |
| 29 | \( 1 + 2.60T + 29T^{2} \) |
| 31 | \( 1 + 10.6T + 31T^{2} \) |
| 37 | \( 1 - 8.73T + 37T^{2} \) |
| 41 | \( 1 + 5.76T + 41T^{2} \) |
| 43 | \( 1 + 9.19T + 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 + 5.27T + 53T^{2} \) |
| 59 | \( 1 + 0.157T + 59T^{2} \) |
| 61 | \( 1 + 4.85T + 61T^{2} \) |
| 67 | \( 1 + 11.0T + 67T^{2} \) |
| 71 | \( 1 - 2.90T + 71T^{2} \) |
| 73 | \( 1 - 10.5T + 73T^{2} \) |
| 79 | \( 1 + 2.80T + 79T^{2} \) |
| 83 | \( 1 - 2.59T + 83T^{2} \) |
| 89 | \( 1 + 4.62T + 89T^{2} \) |
| 97 | \( 1 - 0.747T + 97T^{2} \) |
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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
−8.076629962213235137489999433709, −7.51755009702852327564155201665, −6.64911800668156226800648389352, −5.90497787761715098165888230017, −5.22837403714865297813496029094, −4.26386736046274840430304321475, −3.55333409602864190858839816929, −1.94412963560462591974312452993, −1.30653567426572622043719663877, 0,
1.30653567426572622043719663877, 1.94412963560462591974312452993, 3.55333409602864190858839816929, 4.26386736046274840430304321475, 5.22837403714865297813496029094, 5.90497787761715098165888230017, 6.64911800668156226800648389352, 7.51755009702852327564155201665, 8.076629962213235137489999433709
