| L(s) = 1 | − 1.02e3·2-s − 1.76e5·3-s + 1.04e6·4-s + 1.80e8·6-s + 1.28e9·7-s − 1.07e9·8-s + 2.07e10·9-s − 8.21e9·11-s − 1.85e11·12-s + 2.70e11·13-s − 1.32e12·14-s + 1.09e12·16-s + 1.05e13·17-s − 2.12e13·18-s − 2.60e13·19-s − 2.27e14·21-s + 8.41e12·22-s − 3.27e14·23-s + 1.89e14·24-s − 2.76e14·26-s − 1.81e15·27-s + 1.35e15·28-s + 2.01e15·29-s − 7.82e15·31-s − 1.12e15·32-s + 1.45e15·33-s − 1.08e16·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.72·3-s + 0.5·4-s + 1.22·6-s + 1.72·7-s − 0.353·8-s + 1.98·9-s − 0.0954·11-s − 0.863·12-s + 0.543·13-s − 1.22·14-s + 0.250·16-s + 1.27·17-s − 1.40·18-s − 0.973·19-s − 2.98·21-s + 0.0675·22-s − 1.64·23-s + 0.610·24-s − 0.384·26-s − 1.69·27-s + 0.862·28-s + 0.890·29-s − 1.71·31-s − 0.176·32-s + 0.164·33-s − 0.901·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.02e3T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 1.76e5T + 1.04e10T^{2} \) |
| 7 | \( 1 - 1.28e9T + 5.58e17T^{2} \) |
| 11 | \( 1 + 8.21e9T + 7.40e21T^{2} \) |
| 13 | \( 1 - 2.70e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 1.05e13T + 6.90e25T^{2} \) |
| 19 | \( 1 + 2.60e13T + 7.14e26T^{2} \) |
| 23 | \( 1 + 3.27e14T + 3.94e28T^{2} \) |
| 29 | \( 1 - 2.01e15T + 5.13e30T^{2} \) |
| 31 | \( 1 + 7.82e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 2.12e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + 9.00e16T + 7.38e33T^{2} \) |
| 43 | \( 1 + 6.03e16T + 2.00e34T^{2} \) |
| 47 | \( 1 - 3.50e17T + 1.30e35T^{2} \) |
| 53 | \( 1 - 1.24e18T + 1.62e36T^{2} \) |
| 59 | \( 1 + 5.56e18T + 1.54e37T^{2} \) |
| 61 | \( 1 + 5.81e18T + 3.10e37T^{2} \) |
| 67 | \( 1 - 2.21e19T + 2.22e38T^{2} \) |
| 71 | \( 1 - 3.61e19T + 7.52e38T^{2} \) |
| 73 | \( 1 + 4.54e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + 9.81e19T + 7.08e39T^{2} \) |
| 83 | \( 1 + 4.19e19T + 1.99e40T^{2} \) |
| 89 | \( 1 + 3.94e20T + 8.65e40T^{2} \) |
| 97 | \( 1 + 6.34e20T + 5.27e41T^{2} \) |
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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
−10.88839684504190372903970442698, −10.14445083222343657189088324465, −8.405348027340301702931076139309, −7.44256496919471901821185147194, −6.09484323027338010578384412808, −5.29128998723217115911575277606, −4.14880877057248408840427733886, −1.85542861813644868843119807596, −1.10964839751404106077221278193, 0,
1.10964839751404106077221278193, 1.85542861813644868843119807596, 4.14880877057248408840427733886, 5.29128998723217115911575277606, 6.09484323027338010578384412808, 7.44256496919471901821185147194, 8.405348027340301702931076139309, 10.14445083222343657189088324465, 10.88839684504190372903970442698
