| L(s) = 1 | − 2.35e4·2-s + 4.78e6·3-s + 1.79e7·4-s + 8.07e9·5-s − 1.12e11·6-s − 6.78e11·7-s + 1.22e13·8-s + 2.28e13·9-s − 1.90e14·10-s + 1.41e15·11-s + 8.58e13·12-s + 2.01e15·13-s + 1.59e16·14-s + 3.86e16·15-s − 2.97e17·16-s + 7.13e17·17-s − 5.38e17·18-s + 1.22e18·19-s + 1.44e17·20-s − 3.24e18·21-s − 3.33e19·22-s + 1.51e19·23-s + 5.84e19·24-s − 1.21e20·25-s − 4.74e19·26-s + 1.09e20·27-s − 1.21e19·28-s + ⋯ |
| L(s) = 1 | − 1.01·2-s + 0.577·3-s + 0.0334·4-s + 0.591·5-s − 0.586·6-s − 0.377·7-s + 0.982·8-s + 0.333·9-s − 0.601·10-s + 1.12·11-s + 0.0193·12-s + 0.141·13-s + 0.384·14-s + 0.341·15-s − 1.03·16-s + 1.02·17-s − 0.338·18-s + 0.351·19-s + 0.0197·20-s − 0.218·21-s − 1.14·22-s + 0.271·23-s + 0.567·24-s − 0.650·25-s − 0.144·26-s + 0.192·27-s − 0.0126·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(15)\) |
\(\approx\) |
\(1.944387930\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.944387930\) |
| \(L(\frac{31}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 4.78e6T \) |
| 7 | \( 1 + 6.78e11T \) |
| good | 2 | \( 1 + 2.35e4T + 5.36e8T^{2} \) |
| 5 | \( 1 - 8.07e9T + 1.86e20T^{2} \) |
| 11 | \( 1 - 1.41e15T + 1.58e30T^{2} \) |
| 13 | \( 1 - 2.01e15T + 2.01e32T^{2} \) |
| 17 | \( 1 - 7.13e17T + 4.81e35T^{2} \) |
| 19 | \( 1 - 1.22e18T + 1.21e37T^{2} \) |
| 23 | \( 1 - 1.51e19T + 3.09e39T^{2} \) |
| 29 | \( 1 - 3.20e21T + 2.56e42T^{2} \) |
| 31 | \( 1 - 3.10e21T + 1.77e43T^{2} \) |
| 37 | \( 1 + 4.11e22T + 3.00e45T^{2} \) |
| 41 | \( 1 - 4.12e23T + 5.89e46T^{2} \) |
| 43 | \( 1 + 6.17e23T + 2.34e47T^{2} \) |
| 47 | \( 1 - 2.08e24T + 3.09e48T^{2} \) |
| 53 | \( 1 + 1.67e25T + 1.00e50T^{2} \) |
| 59 | \( 1 - 2.25e25T + 2.26e51T^{2} \) |
| 61 | \( 1 + 1.31e26T + 5.95e51T^{2} \) |
| 67 | \( 1 + 4.36e26T + 9.04e52T^{2} \) |
| 71 | \( 1 + 9.78e26T + 4.85e53T^{2} \) |
| 73 | \( 1 + 6.94e26T + 1.08e54T^{2} \) |
| 79 | \( 1 - 3.14e27T + 1.07e55T^{2} \) |
| 83 | \( 1 - 4.20e26T + 4.50e55T^{2} \) |
| 89 | \( 1 - 8.66e27T + 3.40e56T^{2} \) |
| 97 | \( 1 - 1.46e28T + 4.13e57T^{2} \) |
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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
−12.09014007902107749579714997599, −10.35958831213464377567042719127, −9.559543806236942185683713592260, −8.719858815683540058683194750610, −7.53592090306048151583206909972, −6.21446560929712672028638403175, −4.50801922353353653263814382693, −3.12512365630659253165819469344, −1.63250595822814666792334531754, −0.818188363007793322874071718406,
0.818188363007793322874071718406, 1.63250595822814666792334531754, 3.12512365630659253165819469344, 4.50801922353353653263814382693, 6.21446560929712672028638403175, 7.53592090306048151583206909972, 8.719858815683540058683194750610, 9.559543806236942185683713592260, 10.35958831213464377567042719127, 12.09014007902107749579714997599
