| L(s) = 1 | − 2.24e4·3-s + 1.95e6·5-s + 8.55e7·7-s − 6.59e8·9-s − 1.06e10·11-s + 2.87e9·13-s − 4.37e10·15-s + 1.15e11·17-s − 6.20e11·19-s − 1.91e12·21-s − 1.39e13·23-s + 3.81e12·25-s + 4.08e13·27-s − 2.09e13·29-s + 6.89e13·31-s + 2.38e14·33-s + 1.66e14·35-s + 1.76e14·37-s − 6.45e13·39-s + 3.64e15·41-s − 1.21e15·43-s − 1.28e15·45-s − 8.81e15·47-s − 4.08e15·49-s − 2.58e15·51-s + 3.00e16·53-s − 2.07e16·55-s + ⋯ |
| L(s) = 1 | − 0.657·3-s + 0.447·5-s + 0.800·7-s − 0.567·9-s − 1.35·11-s + 0.0753·13-s − 0.294·15-s + 0.235·17-s − 0.441·19-s − 0.526·21-s − 1.61·23-s + 0.199·25-s + 1.03·27-s − 0.268·29-s + 0.468·31-s + 0.892·33-s + 0.358·35-s + 0.223·37-s − 0.0495·39-s + 1.73·41-s − 0.368·43-s − 0.253·45-s − 1.14·47-s − 0.358·49-s − 0.154·51-s + 1.25·53-s − 0.607·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(\approx\) |
\(1.199345849\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.199345849\) |
| \(L(\frac{21}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 1.95e6T \) |
| good | 3 | \( 1 + 2.24e4T + 1.16e9T^{2} \) |
| 7 | \( 1 - 8.55e7T + 1.13e16T^{2} \) |
| 11 | \( 1 + 1.06e10T + 6.11e19T^{2} \) |
| 13 | \( 1 - 2.87e9T + 1.46e21T^{2} \) |
| 17 | \( 1 - 1.15e11T + 2.39e23T^{2} \) |
| 19 | \( 1 + 6.20e11T + 1.97e24T^{2} \) |
| 23 | \( 1 + 1.39e13T + 7.46e25T^{2} \) |
| 29 | \( 1 + 2.09e13T + 6.10e27T^{2} \) |
| 31 | \( 1 - 6.89e13T + 2.16e28T^{2} \) |
| 37 | \( 1 - 1.76e14T + 6.24e29T^{2} \) |
| 41 | \( 1 - 3.64e15T + 4.39e30T^{2} \) |
| 43 | \( 1 + 1.21e15T + 1.08e31T^{2} \) |
| 47 | \( 1 + 8.81e15T + 5.88e31T^{2} \) |
| 53 | \( 1 - 3.00e16T + 5.77e32T^{2} \) |
| 59 | \( 1 + 9.02e16T + 4.42e33T^{2} \) |
| 61 | \( 1 - 7.62e16T + 8.34e33T^{2} \) |
| 67 | \( 1 + 1.30e17T + 4.95e34T^{2} \) |
| 71 | \( 1 + 3.87e17T + 1.49e35T^{2} \) |
| 73 | \( 1 + 4.19e17T + 2.53e35T^{2} \) |
| 79 | \( 1 - 1.12e18T + 1.13e36T^{2} \) |
| 83 | \( 1 + 9.31e17T + 2.90e36T^{2} \) |
| 89 | \( 1 + 4.12e18T + 1.09e37T^{2} \) |
| 97 | \( 1 + 1.26e19T + 5.60e37T^{2} \) |
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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
−10.83006642198103184410281647656, −9.982338026120589006079284348903, −8.503726162844604164201176620834, −7.68392980778237728333191631494, −6.14598252471898704238009399623, −5.44506096976628523701304534811, −4.45565167658879424024078159249, −2.81671994923582432279146668050, −1.81789185138631342808024553022, −0.46992366820067525624628089440,
0.46992366820067525624628089440, 1.81789185138631342808024553022, 2.81671994923582432279146668050, 4.45565167658879424024078159249, 5.44506096976628523701304534811, 6.14598252471898704238009399623, 7.68392980778237728333191631494, 8.503726162844604164201176620834, 9.982338026120589006079284348903, 10.83006642198103184410281647656
