| L(s) = 1 | − 3.18e3·2-s + 1.98e5·3-s + 1.75e6·4-s − 1.30e8·5-s − 6.31e8·6-s + 2.11e10·8-s − 5.48e10·9-s + 4.16e11·10-s − 8.54e11·11-s + 3.47e11·12-s + 4.39e12·13-s − 2.59e13·15-s − 8.19e13·16-s + 5.22e13·17-s + 1.74e14·18-s + 7.03e14·19-s − 2.29e14·20-s + 2.72e15·22-s − 6.02e15·23-s + 4.18e15·24-s + 5.19e15·25-s − 1.39e16·26-s − 2.95e16·27-s + 2.49e16·29-s + 8.26e16·30-s − 1.34e17·31-s + 8.38e16·32-s + ⋯ |
| L(s) = 1 | − 1.09·2-s + 0.646·3-s + 0.208·4-s − 1.19·5-s − 0.710·6-s + 0.869·8-s − 0.582·9-s + 1.31·10-s − 0.902·11-s + 0.135·12-s + 0.680·13-s − 0.774·15-s − 1.16·16-s + 0.369·17-s + 0.640·18-s + 1.38·19-s − 0.250·20-s + 0.992·22-s − 1.31·23-s + 0.562·24-s + 0.436·25-s − 0.748·26-s − 1.02·27-s + 0.379·29-s + 0.851·30-s − 0.952·31-s + 0.411·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(12)\) |
\(\approx\) |
\(0.3685992066\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3685992066\) |
| \(L(\frac{25}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 + 3.18e3T + 8.38e6T^{2} \) |
| 3 | \( 1 - 1.98e5T + 9.41e10T^{2} \) |
| 5 | \( 1 + 1.30e8T + 1.19e16T^{2} \) |
| 11 | \( 1 + 8.54e11T + 8.95e23T^{2} \) |
| 13 | \( 1 - 4.39e12T + 4.17e25T^{2} \) |
| 17 | \( 1 - 5.22e13T + 1.99e28T^{2} \) |
| 19 | \( 1 - 7.03e14T + 2.57e29T^{2} \) |
| 23 | \( 1 + 6.02e15T + 2.08e31T^{2} \) |
| 29 | \( 1 - 2.49e16T + 4.31e33T^{2} \) |
| 31 | \( 1 + 1.34e17T + 2.00e34T^{2} \) |
| 37 | \( 1 + 1.90e18T + 1.17e36T^{2} \) |
| 41 | \( 1 - 2.84e18T + 1.24e37T^{2} \) |
| 43 | \( 1 - 1.57e17T + 3.71e37T^{2} \) |
| 47 | \( 1 + 1.56e19T + 2.87e38T^{2} \) |
| 53 | \( 1 + 1.88e19T + 4.55e39T^{2} \) |
| 59 | \( 1 - 1.27e20T + 5.36e40T^{2} \) |
| 61 | \( 1 - 1.22e20T + 1.15e41T^{2} \) |
| 67 | \( 1 + 1.57e21T + 9.99e41T^{2} \) |
| 71 | \( 1 - 2.89e20T + 3.79e42T^{2} \) |
| 73 | \( 1 + 1.44e21T + 7.18e42T^{2} \) |
| 79 | \( 1 + 1.25e22T + 4.42e43T^{2} \) |
| 83 | \( 1 - 4.81e21T + 1.37e44T^{2} \) |
| 89 | \( 1 + 2.54e22T + 6.85e44T^{2} \) |
| 97 | \( 1 + 1.26e23T + 4.96e45T^{2} \) |
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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
−10.96107824742911601170175129793, −9.840111591408455883098689176504, −8.685676876765432641738017001391, −8.019429954075773501911553587780, −7.36827061862471136295716531605, −5.45672384637850012665201496128, −3.99523067832619621882499600091, −3.03847666377473385442032389589, −1.60358608560525346198884222459, −0.30699689268574298896377435463,
0.30699689268574298896377435463, 1.60358608560525346198884222459, 3.03847666377473385442032389589, 3.99523067832619621882499600091, 5.45672384637850012665201496128, 7.36827061862471136295716531605, 8.019429954075773501911553587780, 8.685676876765432641738017001391, 9.840111591408455883098689176504, 10.96107824742911601170175129793
