| L(s) = 1 | − 2.58e6·2-s + 4.47e12·4-s − 2.55e14·5-s + 1.98e17·7-s − 5.88e18·8-s + 6.60e20·10-s + 2.76e21·11-s − 8.56e22·13-s − 5.14e23·14-s + 5.36e24·16-s + 3.38e24·17-s + 2.99e26·19-s − 1.14e27·20-s − 7.14e27·22-s + 1.40e28·23-s + 1.98e28·25-s + 2.21e29·26-s + 8.90e29·28-s + 4.94e28·29-s + 5.76e30·31-s − 9.15e29·32-s − 8.75e30·34-s − 5.08e31·35-s − 1.16e32·37-s − 7.73e32·38-s + 1.50e33·40-s − 1.48e32·41-s + ⋯ |
| L(s) = 1 | − 1.74·2-s + 2.03·4-s − 1.19·5-s + 0.942·7-s − 1.80·8-s + 2.08·10-s + 1.23·11-s − 1.24·13-s − 1.64·14-s + 1.10·16-s + 0.202·17-s + 1.82·19-s − 2.43·20-s − 2.15·22-s + 1.71·23-s + 0.436·25-s + 2.17·26-s + 1.91·28-s + 0.0518·29-s + 1.54·31-s − 0.127·32-s − 0.352·34-s − 1.12·35-s − 0.830·37-s − 3.18·38-s + 2.16·40-s − 0.128·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(42-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+41/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(21)\) |
\(\approx\) |
\(0.8740255761\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8740255761\) |
| \(L(\frac{43}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 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
−11.82519394767189630601892378320, −11.47393114554063360796199482444, −9.907433188524873019517225758322, −8.761796951572383544155809845811, −7.69547013582766612117479682604, −6.99702201552922234511984862425, −4.78392744388453468911577115156, −3.06988901815478806955305517106, −1.45871595840921031519091056326, −0.65624621430297348650834640855,
0.65624621430297348650834640855, 1.45871595840921031519091056326, 3.06988901815478806955305517106, 4.78392744388453468911577115156, 6.99702201552922234511984862425, 7.69547013582766612117479682604, 8.761796951572383544155809845811, 9.907433188524873019517225758322, 11.47393114554063360796199482444, 11.82519394767189630601892378320
