| L(s) = 1 | + 1.02e3·2-s + 1.45e5·3-s + 1.04e6·4-s − 9.76e6·5-s + 1.49e8·6-s − 2.36e8·7-s + 1.07e9·8-s + 1.08e10·9-s − 1.00e10·10-s + 1.49e11·11-s + 1.52e11·12-s + 4.89e11·13-s − 2.41e11·14-s − 1.42e12·15-s + 1.09e12·16-s + 5.46e12·17-s + 1.10e13·18-s − 1.12e13·19-s − 1.02e13·20-s − 3.44e13·21-s + 1.52e14·22-s + 1.78e13·23-s + 1.56e14·24-s + 9.53e13·25-s + 5.01e14·26-s + 4.99e13·27-s − 2.47e14·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.42·3-s + 0.5·4-s − 0.447·5-s + 1.00·6-s − 0.315·7-s + 0.353·8-s + 1.03·9-s − 0.316·10-s + 1.73·11-s + 0.712·12-s + 0.984·13-s − 0.223·14-s − 0.637·15-s + 0.250·16-s + 0.657·17-s + 0.730·18-s − 0.419·19-s − 0.223·20-s − 0.450·21-s + 1.22·22-s + 0.0899·23-s + 0.504·24-s + 0.199·25-s + 0.696·26-s + 0.0467·27-s − 0.157·28-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
| \(L(11)\) |
\(\approx\) |
\(5.044118814\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.044118814\) |
| \(L(\frac{23}{2})\) |
|
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 | \( 1 - 1.45e5T + 1.04e10T^{2} \) |
| 7 | \( 1 + 2.36e8T + 5.58e17T^{2} \) |
| 11 | \( 1 - 1.49e11T + 7.40e21T^{2} \) |
| 13 | \( 1 - 4.89e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 5.46e12T + 6.90e25T^{2} \) |
| 19 | \( 1 + 1.12e13T + 7.14e26T^{2} \) |
| 23 | \( 1 - 1.78e13T + 3.94e28T^{2} \) |
| 29 | \( 1 + 2.82e15T + 5.13e30T^{2} \) |
| 31 | \( 1 - 4.24e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 5.74e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + 3.51e16T + 7.38e33T^{2} \) |
| 43 | \( 1 + 2.62e17T + 2.00e34T^{2} \) |
| 47 | \( 1 - 1.08e17T + 1.30e35T^{2} \) |
| 53 | \( 1 + 2.47e18T + 1.62e36T^{2} \) |
| 59 | \( 1 - 2.14e17T + 1.54e37T^{2} \) |
| 61 | \( 1 - 3.64e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 1.13e19T + 2.22e38T^{2} \) |
| 71 | \( 1 - 2.44e19T + 7.52e38T^{2} \) |
| 73 | \( 1 + 4.12e19T + 1.34e39T^{2} \) |
| 79 | \( 1 - 6.83e19T + 7.08e39T^{2} \) |
| 83 | \( 1 - 1.39e20T + 1.99e40T^{2} \) |
| 89 | \( 1 + 4.04e20T + 8.65e40T^{2} \) |
| 97 | \( 1 + 9.18e20T + 5.27e41T^{2} \) |
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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
−15.13882234462509989118536923004, −14.30191180182851090555111216750, −13.11501084736210432123292800820, −11.53921826035046864923926285742, −9.426914530971558626016586784040, −8.100065386988730858243015776923, −6.45860842729809545368635697469, −4.05255449788773471739179722733, −3.20267462228395308125262242873, −1.48514684380487484315958066096,
1.48514684380487484315958066096, 3.20267462228395308125262242873, 4.05255449788773471739179722733, 6.45860842729809545368635697469, 8.100065386988730858243015776923, 9.426914530971558626016586784040, 11.53921826035046864923926285742, 13.11501084736210432123292800820, 14.30191180182851090555111216750, 15.13882234462509989118536923004
