| L(s) = 1 | + 867.·2-s − 4.68e4·3-s + 2.28e5·4-s − 6.21e6·5-s − 4.06e7·6-s + 1.08e8·7-s − 2.56e8·8-s + 1.03e9·9-s − 5.39e9·10-s + 1.80e9·11-s − 1.06e10·12-s + 2.66e10·13-s + 9.36e10·14-s + 2.91e11·15-s − 3.42e11·16-s − 3.93e11·17-s + 8.99e11·18-s + 1.72e12·19-s − 1.41e12·20-s − 5.06e12·21-s + 1.56e12·22-s + 9.57e12·23-s + 1.20e13·24-s + 1.95e13·25-s + 2.30e13·26-s + 5.89e12·27-s + 2.46e13·28-s + ⋯ |
| L(s) = 1 | + 1.19·2-s − 1.37·3-s + 0.434·4-s − 1.42·5-s − 1.64·6-s + 1.01·7-s − 0.676·8-s + 0.891·9-s − 1.70·10-s + 0.230·11-s − 0.598·12-s + 0.695·13-s + 1.21·14-s + 1.95·15-s − 1.24·16-s − 0.804·17-s + 1.06·18-s + 1.22·19-s − 0.619·20-s − 1.39·21-s + 0.276·22-s + 1.10·23-s + 0.930·24-s + 1.02·25-s + 0.833·26-s + 0.148·27-s + 0.440·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{21}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 47 | \( 1 - 1.11e15T \) |
| good | 2 | \( 1 - 867.T + 5.24e5T^{2} \) |
| 3 | \( 1 + 4.68e4T + 1.16e9T^{2} \) |
| 5 | \( 1 + 6.21e6T + 1.90e13T^{2} \) |
| 7 | \( 1 - 1.08e8T + 1.13e16T^{2} \) |
| 11 | \( 1 - 1.80e9T + 6.11e19T^{2} \) |
| 13 | \( 1 - 2.66e10T + 1.46e21T^{2} \) |
| 17 | \( 1 + 3.93e11T + 2.39e23T^{2} \) |
| 19 | \( 1 - 1.72e12T + 1.97e24T^{2} \) |
| 23 | \( 1 - 9.57e12T + 7.46e25T^{2} \) |
| 29 | \( 1 - 5.53e13T + 6.10e27T^{2} \) |
| 31 | \( 1 + 9.26e13T + 2.16e28T^{2} \) |
| 37 | \( 1 - 6.10e14T + 6.24e29T^{2} \) |
| 41 | \( 1 - 8.08e14T + 4.39e30T^{2} \) |
| 43 | \( 1 + 3.26e15T + 1.08e31T^{2} \) |
| 53 | \( 1 + 3.46e15T + 5.77e32T^{2} \) |
| 59 | \( 1 - 4.77e16T + 4.42e33T^{2} \) |
| 61 | \( 1 + 1.80e17T + 8.34e33T^{2} \) |
| 67 | \( 1 + 2.98e17T + 4.95e34T^{2} \) |
| 71 | \( 1 + 6.99e17T + 1.49e35T^{2} \) |
| 73 | \( 1 + 1.26e17T + 2.53e35T^{2} \) |
| 79 | \( 1 - 1.04e18T + 1.13e36T^{2} \) |
| 83 | \( 1 + 1.69e18T + 2.90e36T^{2} \) |
| 89 | \( 1 - 4.34e18T + 1.09e37T^{2} \) |
| 97 | \( 1 - 2.70e18T + 5.60e37T^{2} \) |
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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
−11.53467181256605323393859633539, −10.95848340323817825170644686144, −8.767647798265339591733556020645, −7.36261528096034282198984402018, −6.13469525653464548568483636273, −4.97055018509526884702985188939, −4.41726062882627915392923103028, −3.22812328649294876128382506709, −1.10404279032342041277899976273, 0,
1.10404279032342041277899976273, 3.22812328649294876128382506709, 4.41726062882627915392923103028, 4.97055018509526884702985188939, 6.13469525653464548568483636273, 7.36261528096034282198984402018, 8.767647798265339591733556020645, 10.95848340323817825170644686144, 11.53467181256605323393859633539
