| L(s) = 1 | − 1.02e3·2-s + 4.79e4·3-s + 1.04e6·4-s − 9.76e6·5-s − 4.90e7·6-s + 6.67e8·7-s − 1.07e9·8-s − 8.16e9·9-s + 1.00e10·10-s − 1.26e11·11-s + 5.02e10·12-s + 9.12e11·13-s − 6.83e11·14-s − 4.68e11·15-s + 1.09e12·16-s + 8.60e12·17-s + 8.35e12·18-s − 6.93e12·19-s − 1.02e13·20-s + 3.20e13·21-s + 1.29e14·22-s − 3.30e14·23-s − 5.14e13·24-s + 9.53e13·25-s − 9.34e14·26-s − 8.92e14·27-s + 7.00e14·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.468·3-s + 0.5·4-s − 0.447·5-s − 0.331·6-s + 0.893·7-s − 0.353·8-s − 0.780·9-s + 0.316·10-s − 1.47·11-s + 0.234·12-s + 1.83·13-s − 0.631·14-s − 0.209·15-s + 0.250·16-s + 1.03·17-s + 0.551·18-s − 0.259·19-s − 0.223·20-s + 0.418·21-s + 1.04·22-s − 1.66·23-s − 0.165·24-s + 0.199·25-s − 1.29·26-s − 0.834·27-s + 0.446·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 9.76e6T \) |
| good | 3 | \( 1 - 4.79e4T + 1.04e10T^{2} \) |
| 7 | \( 1 - 6.67e8T + 5.58e17T^{2} \) |
| 11 | \( 1 + 1.26e11T + 7.40e21T^{2} \) |
| 13 | \( 1 - 9.12e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 8.60e12T + 6.90e25T^{2} \) |
| 19 | \( 1 + 6.93e12T + 7.14e26T^{2} \) |
| 23 | \( 1 + 3.30e14T + 3.94e28T^{2} \) |
| 29 | \( 1 + 3.90e15T + 5.13e30T^{2} \) |
| 31 | \( 1 - 3.32e15T + 2.08e31T^{2} \) |
| 37 | \( 1 + 4.33e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + 9.56e16T + 7.38e33T^{2} \) |
| 43 | \( 1 + 7.81e16T + 2.00e34T^{2} \) |
| 47 | \( 1 + 2.03e17T + 1.30e35T^{2} \) |
| 53 | \( 1 - 1.42e18T + 1.62e36T^{2} \) |
| 59 | \( 1 - 2.28e18T + 1.54e37T^{2} \) |
| 61 | \( 1 + 5.32e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 1.31e19T + 2.22e38T^{2} \) |
| 71 | \( 1 + 2.98e19T + 7.52e38T^{2} \) |
| 73 | \( 1 - 1.37e18T + 1.34e39T^{2} \) |
| 79 | \( 1 - 1.74e19T + 7.08e39T^{2} \) |
| 83 | \( 1 - 3.75e19T + 1.99e40T^{2} \) |
| 89 | \( 1 - 2.43e20T + 8.65e40T^{2} \) |
| 97 | \( 1 + 1.80e20T + 5.27e41T^{2} \) |
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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
−15.12359892667593229874454707570, −13.64200455617642625052055268830, −11.66285262007658734522626524496, −10.48634307925554845713591773938, −8.492003153746732251829754573077, −7.88643975889491217582953307268, −5.64939506760527435706315109686, −3.42967016289538218632564970545, −1.76894170409024154823913311715, 0,
1.76894170409024154823913311715, 3.42967016289538218632564970545, 5.64939506760527435706315109686, 7.88643975889491217582953307268, 8.492003153746732251829754573077, 10.48634307925554845713591773938, 11.66285262007658734522626524496, 13.64200455617642625052055268830, 15.12359892667593229874454707570
