| L(s) = 1 | + 307.·2-s + 6.21e4·3-s − 2.00e6·4-s + 9.76e6·5-s + 1.91e7·6-s + 8.89e8·7-s − 1.26e9·8-s − 6.59e9·9-s + 3.00e9·10-s + 1.34e11·11-s − 1.24e11·12-s + 1.54e11·13-s + 2.73e11·14-s + 6.07e11·15-s + 3.81e12·16-s + 1.18e13·17-s − 2.03e12·18-s + 3.56e13·19-s − 1.95e13·20-s + 5.53e13·21-s + 4.14e13·22-s − 2.33e14·23-s − 7.84e13·24-s + 9.53e13·25-s + 4.74e13·26-s − 1.06e15·27-s − 1.78e15·28-s + ⋯ |
| L(s) = 1 | + 0.212·2-s + 0.607·3-s − 0.954·4-s + 0.447·5-s + 0.129·6-s + 1.19·7-s − 0.415·8-s − 0.630·9-s + 0.0950·10-s + 1.56·11-s − 0.580·12-s + 0.310·13-s + 0.253·14-s + 0.271·15-s + 0.866·16-s + 1.42·17-s − 0.134·18-s + 1.33·19-s − 0.427·20-s + 0.723·21-s + 0.332·22-s − 1.17·23-s − 0.252·24-s + 0.199·25-s + 0.0659·26-s − 0.991·27-s − 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(2.505804621\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.505804621\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 9.76e6T \) |
| good | 2 | \( 1 - 307.T + 2.09e6T^{2} \) |
| 3 | \( 1 - 6.21e4T + 1.04e10T^{2} \) |
| 7 | \( 1 - 8.89e8T + 5.58e17T^{2} \) |
| 11 | \( 1 - 1.34e11T + 7.40e21T^{2} \) |
| 13 | \( 1 - 1.54e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 1.18e13T + 6.90e25T^{2} \) |
| 19 | \( 1 - 3.56e13T + 7.14e26T^{2} \) |
| 23 | \( 1 + 2.33e14T + 3.94e28T^{2} \) |
| 29 | \( 1 - 1.32e15T + 5.13e30T^{2} \) |
| 31 | \( 1 + 5.02e15T + 2.08e31T^{2} \) |
| 37 | \( 1 + 1.85e16T + 8.55e32T^{2} \) |
| 41 | \( 1 - 4.73e16T + 7.38e33T^{2} \) |
| 43 | \( 1 + 1.45e17T + 2.00e34T^{2} \) |
| 47 | \( 1 - 4.04e16T + 1.30e35T^{2} \) |
| 53 | \( 1 - 1.70e17T + 1.62e36T^{2} \) |
| 59 | \( 1 - 1.69e18T + 1.54e37T^{2} \) |
| 61 | \( 1 + 5.24e18T + 3.10e37T^{2} \) |
| 67 | \( 1 - 1.98e19T + 2.22e38T^{2} \) |
| 71 | \( 1 - 2.94e19T + 7.52e38T^{2} \) |
| 73 | \( 1 - 5.41e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + 1.21e20T + 7.08e39T^{2} \) |
| 83 | \( 1 + 1.69e20T + 1.99e40T^{2} \) |
| 89 | \( 1 + 1.96e20T + 8.65e40T^{2} \) |
| 97 | \( 1 - 1.34e21T + 5.27e41T^{2} \) |
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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
−18.29983010359186925568256570894, −17.10195862837365663614904136186, −14.36440880263439952120202197020, −14.10778049868399864945490946320, −11.86518593527625363574269808828, −9.485039857849805393942143528134, −8.189607042302804712651526080125, −5.47162626704009787185637419259, −3.64241485687401878837758512525, −1.32884026326969986671623799461,
1.32884026326969986671623799461, 3.64241485687401878837758512525, 5.47162626704009787185637419259, 8.189607042302804712651526080125, 9.485039857849805393942143528134, 11.86518593527625363574269808828, 14.10778049868399864945490946320, 14.36440880263439952120202197020, 17.10195862837365663614904136186, 18.29983010359186925568256570894
