| L(s) = 1 | + 5.35e4·3-s + 7.60e6·5-s + 1.50e7·7-s + 1.70e9·9-s + 1.11e10·11-s + 4.95e10·13-s + 4.07e11·15-s + 6.36e11·17-s − 1.84e12·19-s + 8.03e11·21-s + 2.63e12·23-s + 3.88e13·25-s + 2.92e13·27-s − 7.77e13·29-s − 1.54e14·31-s + 5.97e14·33-s + 1.14e14·35-s + 7.04e13·37-s + 2.65e15·39-s + 2.37e15·41-s − 5.25e14·43-s + 1.29e16·45-s − 1.03e16·47-s − 1.11e16·49-s + 3.41e16·51-s − 2.20e16·53-s + 8.49e16·55-s + ⋯ |
| L(s) = 1 | + 1.57·3-s + 1.74·5-s + 0.140·7-s + 1.46·9-s + 1.42·11-s + 1.29·13-s + 2.73·15-s + 1.30·17-s − 1.31·19-s + 0.220·21-s + 0.305·23-s + 2.03·25-s + 0.737·27-s − 0.994·29-s − 1.05·31-s + 2.24·33-s + 0.244·35-s + 0.0890·37-s + 2.03·39-s + 1.13·41-s − 0.159·43-s + 2.55·45-s − 1.34·47-s − 0.980·49-s + 2.04·51-s − 0.919·53-s + 2.48·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(\approx\) |
\(7.553065660\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.553065660\) |
| \(L(\frac{21}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 5.35e4T + 1.16e9T^{2} \) |
| 5 | \( 1 - 7.60e6T + 1.90e13T^{2} \) |
| 7 | \( 1 - 1.50e7T + 1.13e16T^{2} \) |
| 11 | \( 1 - 1.11e10T + 6.11e19T^{2} \) |
| 13 | \( 1 - 4.95e10T + 1.46e21T^{2} \) |
| 17 | \( 1 - 6.36e11T + 2.39e23T^{2} \) |
| 19 | \( 1 + 1.84e12T + 1.97e24T^{2} \) |
| 23 | \( 1 - 2.63e12T + 7.46e25T^{2} \) |
| 29 | \( 1 + 7.77e13T + 6.10e27T^{2} \) |
| 31 | \( 1 + 1.54e14T + 2.16e28T^{2} \) |
| 37 | \( 1 - 7.04e13T + 6.24e29T^{2} \) |
| 41 | \( 1 - 2.37e15T + 4.39e30T^{2} \) |
| 43 | \( 1 + 5.25e14T + 1.08e31T^{2} \) |
| 47 | \( 1 + 1.03e16T + 5.88e31T^{2} \) |
| 53 | \( 1 + 2.20e16T + 5.77e32T^{2} \) |
| 59 | \( 1 - 7.48e16T + 4.42e33T^{2} \) |
| 61 | \( 1 + 1.52e17T + 8.34e33T^{2} \) |
| 67 | \( 1 - 1.71e17T + 4.95e34T^{2} \) |
| 71 | \( 1 + 8.23e16T + 1.49e35T^{2} \) |
| 73 | \( 1 + 1.03e17T + 2.53e35T^{2} \) |
| 79 | \( 1 + 3.92e17T + 1.13e36T^{2} \) |
| 83 | \( 1 + 1.61e18T + 2.90e36T^{2} \) |
| 89 | \( 1 + 2.10e17T + 1.09e37T^{2} \) |
| 97 | \( 1 - 7.42e17T + 5.60e37T^{2} \) |
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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
−10.93322034901186687336740511100, −9.629285781776934255656961582196, −9.156493171284434594726981074583, −8.205363072396416510526292357543, −6.68359550045296667885353208736, −5.72957713329666551091490751476, −4.00744036159273760478404151468, −3.03700903623197131669568413424, −1.75897153785617415609894797083, −1.41234555685935786314779980674,
1.41234555685935786314779980674, 1.75897153785617415609894797083, 3.03700903623197131669568413424, 4.00744036159273760478404151468, 5.72957713329666551091490751476, 6.68359550045296667885353208736, 8.205363072396416510526292357543, 9.156493171284434594726981074583, 9.629285781776934255656961582196, 10.93322034901186687336740511100
