| L(s) = 1 | − 2.80e3·2-s − 1.77e5·3-s − 5.15e5·4-s − 1.60e8·5-s + 4.97e8·6-s − 8.06e9·7-s + 2.49e10·8-s + 3.13e10·9-s + 4.50e11·10-s + 4.37e11·11-s + 9.13e10·12-s + 2.40e12·13-s + 2.26e13·14-s + 2.84e13·15-s − 6.57e13·16-s − 7.22e13·17-s − 8.80e13·18-s − 7.65e14·19-s + 8.27e13·20-s + 1.42e15·21-s − 1.22e15·22-s + 5.30e15·23-s − 4.42e15·24-s + 1.38e16·25-s − 6.74e15·26-s − 5.55e15·27-s + 4.15e15·28-s + ⋯ |
| L(s) = 1 | − 0.968·2-s − 0.577·3-s − 0.0614·4-s − 1.46·5-s + 0.559·6-s − 1.54·7-s + 1.02·8-s + 0.333·9-s + 1.42·10-s + 0.462·11-s + 0.0354·12-s + 0.371·13-s + 1.49·14-s + 0.848·15-s − 0.934·16-s − 0.511·17-s − 0.322·18-s − 1.50·19-s + 0.0903·20-s + 0.890·21-s − 0.448·22-s + 1.16·23-s − 0.593·24-s + 1.15·25-s − 0.360·26-s − 0.192·27-s + 0.0947·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(12)\) |
\(\approx\) |
\(0.2451422364\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2451422364\) |
| \(L(\frac{25}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 1.77e5T \) |
| good | 2 | \( 1 + 2.80e3T + 8.38e6T^{2} \) |
| 5 | \( 1 + 1.60e8T + 1.19e16T^{2} \) |
| 7 | \( 1 + 8.06e9T + 2.73e19T^{2} \) |
| 11 | \( 1 - 4.37e11T + 8.95e23T^{2} \) |
| 13 | \( 1 - 2.40e12T + 4.17e25T^{2} \) |
| 17 | \( 1 + 7.22e13T + 1.99e28T^{2} \) |
| 19 | \( 1 + 7.65e14T + 2.57e29T^{2} \) |
| 23 | \( 1 - 5.30e15T + 2.08e31T^{2} \) |
| 29 | \( 1 - 3.30e16T + 4.31e33T^{2} \) |
| 31 | \( 1 + 1.91e17T + 2.00e34T^{2} \) |
| 37 | \( 1 + 1.19e18T + 1.17e36T^{2} \) |
| 41 | \( 1 + 1.08e18T + 1.24e37T^{2} \) |
| 43 | \( 1 - 9.41e18T + 3.71e37T^{2} \) |
| 47 | \( 1 - 1.46e19T + 2.87e38T^{2} \) |
| 53 | \( 1 - 3.45e19T + 4.55e39T^{2} \) |
| 59 | \( 1 + 2.65e20T + 5.36e40T^{2} \) |
| 61 | \( 1 - 5.20e20T + 1.15e41T^{2} \) |
| 67 | \( 1 + 3.34e20T + 9.99e41T^{2} \) |
| 71 | \( 1 - 1.99e21T + 3.79e42T^{2} \) |
| 73 | \( 1 - 2.19e20T + 7.18e42T^{2} \) |
| 79 | \( 1 + 3.54e20T + 4.42e43T^{2} \) |
| 83 | \( 1 + 4.81e21T + 1.37e44T^{2} \) |
| 89 | \( 1 + 3.80e22T + 6.85e44T^{2} \) |
| 97 | \( 1 - 9.60e22T + 4.96e45T^{2} \) |
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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
−19.62731892573877575492218507638, −18.90479510777121456839538943446, −16.92077067997102888633418088038, −15.72907926773758392455312018680, −12.73113147297691245499278761342, −10.80499844067584320318427704000, −8.901077629976462094078466883187, −6.95262491792968601961422457546, −3.97606202658624578829462454138, −0.46052755934266658061753194988,
0.46052755934266658061753194988, 3.97606202658624578829462454138, 6.95262491792968601961422457546, 8.901077629976462094078466883187, 10.80499844067584320318427704000, 12.73113147297691245499278761342, 15.72907926773758392455312018680, 16.92077067997102888633418088038, 18.90479510777121456839538943446, 19.62731892573877575492218507638
