| L(s) = 1 | − 1.63e4·3-s − 6.88e7·7-s − 8.95e8·9-s + 2.10e9·11-s − 2.82e10·13-s + 4.97e11·17-s − 9.50e11·19-s + 1.12e12·21-s + 4.43e11·23-s + 3.36e13·27-s + 8.85e13·29-s − 4.77e13·31-s − 3.43e13·33-s + 3.94e14·37-s + 4.61e14·39-s − 1.34e15·41-s + 4.89e15·43-s + 4.56e15·47-s − 6.65e15·49-s − 8.12e15·51-s + 1.34e16·53-s + 1.55e16·57-s − 9.95e16·59-s − 1.00e17·61-s + 6.16e16·63-s + 3.34e17·67-s − 7.25e15·69-s + ⋯ |
| L(s) = 1 | − 0.479·3-s − 0.645·7-s − 0.770·9-s + 0.268·11-s − 0.739·13-s + 1.01·17-s − 0.675·19-s + 0.309·21-s + 0.0513·23-s + 0.848·27-s + 1.13·29-s − 0.324·31-s − 0.128·33-s + 0.499·37-s + 0.354·39-s − 0.642·41-s + 1.48·43-s + 0.595·47-s − 0.583·49-s − 0.487·51-s + 0.561·53-s + 0.324·57-s − 1.49·59-s − 1.10·61-s + 0.497·63-s + 1.50·67-s − 0.0246·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 1.63e4T + 1.16e9T^{2} \) |
| 7 | \( 1 + 6.88e7T + 1.13e16T^{2} \) |
| 11 | \( 1 - 2.10e9T + 6.11e19T^{2} \) |
| 13 | \( 1 + 2.82e10T + 1.46e21T^{2} \) |
| 17 | \( 1 - 4.97e11T + 2.39e23T^{2} \) |
| 19 | \( 1 + 9.50e11T + 1.97e24T^{2} \) |
| 23 | \( 1 - 4.43e11T + 7.46e25T^{2} \) |
| 29 | \( 1 - 8.85e13T + 6.10e27T^{2} \) |
| 31 | \( 1 + 4.77e13T + 2.16e28T^{2} \) |
| 37 | \( 1 - 3.94e14T + 6.24e29T^{2} \) |
| 41 | \( 1 + 1.34e15T + 4.39e30T^{2} \) |
| 43 | \( 1 - 4.89e15T + 1.08e31T^{2} \) |
| 47 | \( 1 - 4.56e15T + 5.88e31T^{2} \) |
| 53 | \( 1 - 1.34e16T + 5.77e32T^{2} \) |
| 59 | \( 1 + 9.95e16T + 4.42e33T^{2} \) |
| 61 | \( 1 + 1.00e17T + 8.34e33T^{2} \) |
| 67 | \( 1 - 3.34e17T + 4.95e34T^{2} \) |
| 71 | \( 1 - 3.02e17T + 1.49e35T^{2} \) |
| 73 | \( 1 + 1.56e16T + 2.53e35T^{2} \) |
| 79 | \( 1 + 5.08e17T + 1.13e36T^{2} \) |
| 83 | \( 1 + 3.81e17T + 2.90e36T^{2} \) |
| 89 | \( 1 - 4.13e18T + 1.09e37T^{2} \) |
| 97 | \( 1 - 1.49e18T + 5.60e37T^{2} \) |
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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
−9.951658091608680276914345400502, −8.938075132461071736447992157276, −7.77188652611363469862921382914, −6.57441458902675110060462858281, −5.75932043777888106983445993773, −4.68186540551600056201150029621, −3.36918991032507241699341242758, −2.41575166251610749931414545308, −0.935603515249114998917841005612, 0,
0.935603515249114998917841005612, 2.41575166251610749931414545308, 3.36918991032507241699341242758, 4.68186540551600056201150029621, 5.75932043777888106983445993773, 6.57441458902675110060462858281, 7.77188652611363469862921382914, 8.938075132461071736447992157276, 9.951658091608680276914345400502
