| L(s) = 1 | − 1.77e5·3-s + 1.22e8·5-s + 4.92e9·7-s + 3.13e10·9-s − 1.05e11·11-s − 4.97e11·13-s − 2.17e13·15-s − 3.26e13·17-s + 7.69e14·19-s − 8.72e14·21-s + 5.13e15·23-s + 3.11e15·25-s − 5.55e15·27-s + 7.47e16·29-s + 1.28e17·31-s + 1.86e16·33-s + 6.04e17·35-s − 8.87e17·37-s + 8.81e16·39-s + 2.40e18·41-s + 5.95e18·43-s + 3.84e18·45-s − 2.36e19·47-s − 3.09e18·49-s + 5.77e18·51-s + 3.46e19·53-s − 1.28e19·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.12·5-s + 0.941·7-s + 0.333·9-s − 0.111·11-s − 0.0770·13-s − 0.648·15-s − 0.230·17-s + 1.51·19-s − 0.543·21-s + 1.12·23-s + 0.261·25-s − 0.192·27-s + 1.13·29-s + 0.907·31-s + 0.0641·33-s + 1.05·35-s − 0.820·37-s + 0.0444·39-s + 0.682·41-s + 0.977·43-s + 0.374·45-s − 1.39·47-s − 0.113·49-s + 0.133·51-s + 0.513·53-s − 0.124·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(12)\) |
\(\approx\) |
\(3.089556085\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.089556085\) |
| \(L(\frac{25}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.77e5T \) |
| good | 5 | \( 1 - 1.22e8T + 1.19e16T^{2} \) |
| 7 | \( 1 - 4.92e9T + 2.73e19T^{2} \) |
| 11 | \( 1 + 1.05e11T + 8.95e23T^{2} \) |
| 13 | \( 1 + 4.97e11T + 4.17e25T^{2} \) |
| 17 | \( 1 + 3.26e13T + 1.99e28T^{2} \) |
| 19 | \( 1 - 7.69e14T + 2.57e29T^{2} \) |
| 23 | \( 1 - 5.13e15T + 2.08e31T^{2} \) |
| 29 | \( 1 - 7.47e16T + 4.31e33T^{2} \) |
| 31 | \( 1 - 1.28e17T + 2.00e34T^{2} \) |
| 37 | \( 1 + 8.87e17T + 1.17e36T^{2} \) |
| 41 | \( 1 - 2.40e18T + 1.24e37T^{2} \) |
| 43 | \( 1 - 5.95e18T + 3.71e37T^{2} \) |
| 47 | \( 1 + 2.36e19T + 2.87e38T^{2} \) |
| 53 | \( 1 - 3.46e19T + 4.55e39T^{2} \) |
| 59 | \( 1 + 8.87e19T + 5.36e40T^{2} \) |
| 61 | \( 1 + 4.63e20T + 1.15e41T^{2} \) |
| 67 | \( 1 + 6.77e20T + 9.99e41T^{2} \) |
| 71 | \( 1 + 1.80e19T + 3.79e42T^{2} \) |
| 73 | \( 1 + 8.89e20T + 7.18e42T^{2} \) |
| 79 | \( 1 - 9.36e21T + 4.42e43T^{2} \) |
| 83 | \( 1 - 1.60e22T + 1.37e44T^{2} \) |
| 89 | \( 1 + 1.20e22T + 6.85e44T^{2} \) |
| 97 | \( 1 + 7.53e22T + 4.96e45T^{2} \) |
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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
−11.15767366228957422292025463706, −10.15245619591848495899241036559, −9.123891755317782052183389778009, −7.73890681429498179970622162894, −6.50031987077967646542957695373, −5.40502634381703444341267623531, −4.68534984962896161180718854281, −2.91163846043467559416875173345, −1.67042629507672000974017199352, −0.854986226075154746468411020697,
0.854986226075154746468411020697, 1.67042629507672000974017199352, 2.91163846043467559416875173345, 4.68534984962896161180718854281, 5.40502634381703444341267623531, 6.50031987077967646542957695373, 7.73890681429498179970622162894, 9.123891755317782052183389778009, 10.15245619591848495899241036559, 11.15767366228957422292025463706
