| L(s) = 1 | − 1.02e3·2-s − 1.74e5·3-s + 1.04e6·4-s − 9.76e6·5-s + 1.78e8·6-s − 9.60e8·7-s − 1.07e9·8-s + 2.00e10·9-s + 1.00e10·10-s + 8.54e10·11-s − 1.83e11·12-s + 9.74e11·13-s + 9.83e11·14-s + 1.70e12·15-s + 1.09e12·16-s − 1.17e13·17-s − 2.05e13·18-s + 1.41e13·19-s − 1.02e13·20-s + 1.67e14·21-s − 8.75e13·22-s − 2.65e13·23-s + 1.87e14·24-s + 9.53e13·25-s − 9.97e14·26-s − 1.67e15·27-s − 1.00e15·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.70·3-s + 0.5·4-s − 0.447·5-s + 1.20·6-s − 1.28·7-s − 0.353·8-s + 1.91·9-s + 0.316·10-s + 0.993·11-s − 0.853·12-s + 1.96·13-s + 0.908·14-s + 0.763·15-s + 0.250·16-s − 1.40·17-s − 1.35·18-s + 0.528·19-s − 0.223·20-s + 2.19·21-s − 0.702·22-s − 0.133·23-s + 0.603·24-s + 0.199·25-s − 1.38·26-s − 1.56·27-s − 0.642·28-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 | \( 1 + 1.74e5T + 1.04e10T^{2} \) |
| 7 | \( 1 + 9.60e8T + 5.58e17T^{2} \) |
| 11 | \( 1 - 8.54e10T + 7.40e21T^{2} \) |
| 13 | \( 1 - 9.74e11T + 2.47e23T^{2} \) |
| 17 | \( 1 + 1.17e13T + 6.90e25T^{2} \) |
| 19 | \( 1 - 1.41e13T + 7.14e26T^{2} \) |
| 23 | \( 1 + 2.65e13T + 3.94e28T^{2} \) |
| 29 | \( 1 - 1.45e15T + 5.13e30T^{2} \) |
| 31 | \( 1 + 7.63e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 1.09e16T + 8.55e32T^{2} \) |
| 41 | \( 1 - 7.93e16T + 7.38e33T^{2} \) |
| 43 | \( 1 + 8.36e16T + 2.00e34T^{2} \) |
| 47 | \( 1 - 3.57e17T + 1.30e35T^{2} \) |
| 53 | \( 1 + 8.17e17T + 1.62e36T^{2} \) |
| 59 | \( 1 - 8.21e17T + 1.54e37T^{2} \) |
| 61 | \( 1 - 4.53e18T + 3.10e37T^{2} \) |
| 67 | \( 1 - 8.02e18T + 2.22e38T^{2} \) |
| 71 | \( 1 + 5.25e19T + 7.52e38T^{2} \) |
| 73 | \( 1 - 9.28e18T + 1.34e39T^{2} \) |
| 79 | \( 1 - 1.08e19T + 7.08e39T^{2} \) |
| 83 | \( 1 + 4.79e19T + 1.99e40T^{2} \) |
| 89 | \( 1 - 2.35e20T + 8.65e40T^{2} \) |
| 97 | \( 1 - 7.25e20T + 5.27e41T^{2} \) |
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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
−15.85839538305185702401666177148, −12.99776444958657970857434160675, −11.61865778298513851339169783129, −10.73804741737209801406969588700, −9.118563010103317364979725732703, −6.79751258135651795703609932208, −6.04213711772108342139157952173, −3.86191063965913166125107408071, −1.11642112105821876100684601202, 0,
1.11642112105821876100684601202, 3.86191063965913166125107408071, 6.04213711772108342139157952173, 6.79751258135651795703609932208, 9.118563010103317364979725732703, 10.73804741737209801406969588700, 11.61865778298513851339169783129, 12.99776444958657970857434160675, 15.85839538305185702401666177148
