| L(s) = 1 | − 1.05e5·5-s + 5.01e5·7-s − 9.46e7·11-s + 2.61e8·13-s − 2.42e9·17-s + 1.15e9·19-s + 2.23e10·23-s − 1.94e10·25-s − 1.12e11·29-s − 1.67e11·31-s − 5.28e10·35-s + 7.53e11·37-s + 1.42e12·41-s − 1.58e12·43-s + 1.13e12·47-s − 4.49e12·49-s + 5.07e12·53-s + 9.97e12·55-s − 1.15e13·59-s + 1.56e13·61-s − 2.75e13·65-s − 7.82e13·67-s + 6.47e13·71-s + 7.33e13·73-s − 4.75e13·77-s + 1.87e14·79-s + 4.39e13·83-s + ⋯ |
| L(s) = 1 | − 0.602·5-s + 0.230·7-s − 1.46·11-s + 1.15·13-s − 1.43·17-s + 0.296·19-s + 1.37·23-s − 0.636·25-s − 1.21·29-s − 1.09·31-s − 0.138·35-s + 1.30·37-s + 1.14·41-s − 0.889·43-s + 0.325·47-s − 0.946·49-s + 0.593·53-s + 0.883·55-s − 0.603·59-s + 0.638·61-s − 0.696·65-s − 1.57·67-s + 0.844·71-s + 0.776·73-s − 0.337·77-s + 1.09·79-s + 0.177·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(1.368585909\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.368585909\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 1.05e5T + 3.05e10T^{2} \) |
| 7 | \( 1 - 5.01e5T + 4.74e12T^{2} \) |
| 11 | \( 1 + 9.46e7T + 4.17e15T^{2} \) |
| 13 | \( 1 - 2.61e8T + 5.11e16T^{2} \) |
| 17 | \( 1 + 2.42e9T + 2.86e18T^{2} \) |
| 19 | \( 1 - 1.15e9T + 1.51e19T^{2} \) |
| 23 | \( 1 - 2.23e10T + 2.66e20T^{2} \) |
| 29 | \( 1 + 1.12e11T + 8.62e21T^{2} \) |
| 31 | \( 1 + 1.67e11T + 2.34e22T^{2} \) |
| 37 | \( 1 - 7.53e11T + 3.33e23T^{2} \) |
| 41 | \( 1 - 1.42e12T + 1.55e24T^{2} \) |
| 43 | \( 1 + 1.58e12T + 3.17e24T^{2} \) |
| 47 | \( 1 - 1.13e12T + 1.20e25T^{2} \) |
| 53 | \( 1 - 5.07e12T + 7.31e25T^{2} \) |
| 59 | \( 1 + 1.15e13T + 3.65e26T^{2} \) |
| 61 | \( 1 - 1.56e13T + 6.02e26T^{2} \) |
| 67 | \( 1 + 7.82e13T + 2.46e27T^{2} \) |
| 71 | \( 1 - 6.47e13T + 5.87e27T^{2} \) |
| 73 | \( 1 - 7.33e13T + 8.90e27T^{2} \) |
| 79 | \( 1 - 1.87e14T + 2.91e28T^{2} \) |
| 83 | \( 1 - 4.39e13T + 6.11e28T^{2} \) |
| 89 | \( 1 - 5.90e13T + 1.74e29T^{2} \) |
| 97 | \( 1 + 1.06e15T + 6.33e29T^{2} \) |
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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
−11.27046328070072407417719015710, −10.88612537289584657037546977141, −9.279436284619123828045254296139, −8.190599157710977930467694520090, −7.25243252180669694715078416920, −5.80826410816879233639379140719, −4.60724998767307069459915035841, −3.38095214536533657855275535167, −2.06768112601585523464538701572, −0.54731342519786320805850882190,
0.54731342519786320805850882190, 2.06768112601585523464538701572, 3.38095214536533657855275535167, 4.60724998767307069459915035841, 5.80826410816879233639379140719, 7.25243252180669694715078416920, 8.190599157710977930467694520090, 9.279436284619123828045254296139, 10.88612537289584657037546977141, 11.27046328070072407417719015710
