| L(s) = 1 | + 2.58e6·2-s + 3.48e9·3-s + 4.47e12·4-s + 2.55e14·5-s + 9.00e15·6-s + 1.98e17·7-s + 5.88e18·8-s + 1.21e19·9-s + 6.60e20·10-s − 2.76e21·11-s + 1.56e22·12-s − 8.56e22·13-s + 5.14e23·14-s + 8.91e23·15-s + 5.36e24·16-s − 3.38e24·17-s + 3.14e25·18-s + 2.99e26·19-s + 1.14e27·20-s + 6.93e26·21-s − 7.14e27·22-s − 1.40e28·23-s + 2.05e28·24-s + 1.98e28·25-s − 2.21e29·26-s + 4.23e28·27-s + 8.90e29·28-s + ⋯ |
| L(s) = 1 | + 1.74·2-s + 0.577·3-s + 2.03·4-s + 1.19·5-s + 1.00·6-s + 0.942·7-s + 1.80·8-s + 0.333·9-s + 2.08·10-s − 1.23·11-s + 1.17·12-s − 1.24·13-s + 1.64·14-s + 0.691·15-s + 1.10·16-s − 0.202·17-s + 0.580·18-s + 1.82·19-s + 2.43·20-s + 0.544·21-s − 2.15·22-s − 1.71·23-s + 1.04·24-s + 0.436·25-s − 2.17·26-s + 0.192·27-s + 1.91·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(42-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+41/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(21)\) |
\(\approx\) |
\(8.326711406\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.326711406\) |
| \(L(\frac{43}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3.48e9T \) |
| good | 2 | \( 1 - 2.58e6T + 2.19e12T^{2} \) |
| 5 | \( 1 - 2.55e14T + 4.54e28T^{2} \) |
| 7 | \( 1 - 1.98e17T + 4.45e34T^{2} \) |
| 11 | \( 1 + 2.76e21T + 4.97e42T^{2} \) |
| 13 | \( 1 + 8.56e22T + 4.69e45T^{2} \) |
| 17 | \( 1 + 3.38e24T + 2.80e50T^{2} \) |
| 19 | \( 1 - 2.99e26T + 2.68e52T^{2} \) |
| 23 | \( 1 + 1.40e28T + 6.77e55T^{2} \) |
| 29 | \( 1 + 4.94e28T + 9.08e59T^{2} \) |
| 31 | \( 1 - 5.76e30T + 1.39e61T^{2} \) |
| 37 | \( 1 + 1.16e32T + 1.97e64T^{2} \) |
| 41 | \( 1 - 1.48e32T + 1.33e66T^{2} \) |
| 43 | \( 1 - 2.64e32T + 9.38e66T^{2} \) |
| 47 | \( 1 - 2.16e34T + 3.59e68T^{2} \) |
| 53 | \( 1 - 5.57e34T + 4.95e70T^{2} \) |
| 59 | \( 1 + 2.05e36T + 4.02e72T^{2} \) |
| 61 | \( 1 - 1.01e36T + 1.57e73T^{2} \) |
| 67 | \( 1 + 2.42e37T + 7.39e74T^{2} \) |
| 71 | \( 1 - 5.77e36T + 7.97e75T^{2} \) |
| 73 | \( 1 + 1.22e38T + 2.49e76T^{2} \) |
| 79 | \( 1 + 7.70e38T + 6.34e77T^{2} \) |
| 83 | \( 1 - 1.70e39T + 4.81e78T^{2} \) |
| 89 | \( 1 + 4.08e39T + 8.41e79T^{2} \) |
| 97 | \( 1 - 1.60e40T + 2.86e81T^{2} \) |
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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
−15.66299243622715711507632580677, −14.21702509205311263202458446500, −13.60348642219944607458708691625, −12.04123026317238516471208353394, −10.08139757531473401389038677824, −7.57692701435250980805046229326, −5.67464253258325347012894904810, −4.71538140726844446009909166281, −2.80124302747572001597600018569, −1.91314548178943544225527390400,
1.91314548178943544225527390400, 2.80124302747572001597600018569, 4.71538140726844446009909166281, 5.67464253258325347012894904810, 7.57692701435250980805046229326, 10.08139757531473401389038677824, 12.04123026317238516471208353394, 13.60348642219944607458708691625, 14.21702509205311263202458446500, 15.66299243622715711507632580677
