L(s) = 1 | − 1.34e6·2-s − 3.48e9·3-s − 3.91e11·4-s + 1.06e14·5-s + 4.68e15·6-s − 3.99e17·7-s + 3.48e18·8-s + 1.21e19·9-s − 1.43e20·10-s + 2.59e21·11-s + 1.36e21·12-s − 3.34e22·13-s + 5.37e23·14-s − 3.72e23·15-s − 3.82e24·16-s + 2.78e25·17-s − 1.63e25·18-s + 2.51e26·19-s − 4.18e25·20-s + 1.39e27·21-s − 3.49e27·22-s − 2.03e27·23-s − 1.21e28·24-s − 3.40e28·25-s + 4.49e28·26-s − 4.23e28·27-s + 1.56e29·28-s + ⋯ |
L(s) = 1 | − 0.906·2-s − 0.577·3-s − 0.178·4-s + 0.501·5-s + 0.523·6-s − 1.89·7-s + 1.06·8-s + 0.333·9-s − 0.454·10-s + 1.16·11-s + 0.102·12-s − 0.488·13-s + 1.71·14-s − 0.289·15-s − 0.790·16-s + 1.66·17-s − 0.302·18-s + 1.53·19-s − 0.0892·20-s + 1.09·21-s − 1.05·22-s − 0.246·23-s − 0.616·24-s − 0.748·25-s + 0.442·26-s − 0.192·27-s + 0.336·28-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
\(L(21)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{43}{2})\) |
|
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 | \( 1 + 1.34e6T + 2.19e12T^{2} \) |
| 5 | \( 1 - 1.06e14T + 4.54e28T^{2} \) |
| 7 | \( 1 + 3.99e17T + 4.45e34T^{2} \) |
| 11 | \( 1 - 2.59e21T + 4.97e42T^{2} \) |
| 13 | \( 1 + 3.34e22T + 4.69e45T^{2} \) |
| 17 | \( 1 - 2.78e25T + 2.80e50T^{2} \) |
| 19 | \( 1 - 2.51e26T + 2.68e52T^{2} \) |
| 23 | \( 1 + 2.03e27T + 6.77e55T^{2} \) |
| 29 | \( 1 + 6.39e29T + 9.08e59T^{2} \) |
| 31 | \( 1 + 3.99e29T + 1.39e61T^{2} \) |
| 37 | \( 1 + 1.48e32T + 1.97e64T^{2} \) |
| 41 | \( 1 - 5.94e31T + 1.33e66T^{2} \) |
| 43 | \( 1 + 1.25e33T + 9.38e66T^{2} \) |
| 47 | \( 1 - 3.54e33T + 3.59e68T^{2} \) |
| 53 | \( 1 - 1.54e34T + 4.95e70T^{2} \) |
| 59 | \( 1 + 1.46e35T + 4.02e72T^{2} \) |
| 61 | \( 1 + 4.04e36T + 1.57e73T^{2} \) |
| 67 | \( 1 + 8.80e36T + 7.39e74T^{2} \) |
| 71 | \( 1 - 1.53e38T + 7.97e75T^{2} \) |
| 73 | \( 1 + 2.36e38T + 2.49e76T^{2} \) |
| 79 | \( 1 + 4.62e38T + 6.34e77T^{2} \) |
| 83 | \( 1 - 1.09e39T + 4.81e78T^{2} \) |
| 89 | \( 1 + 1.60e40T + 8.41e79T^{2} \) |
| 97 | \( 1 - 8.97e38T + 2.86e81T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.27731134020911919704747936826, −13.78454184469069281023438949660, −12.21530391491336415627324254040, −9.950068406572632759658882952468, −9.463742439339232452374340233725, −7.19649503812756783822299610539, −5.70406383875546182452588870599, −3.54048434342297253000106235990, −1.23714202555919116186724395528, 0,
1.23714202555919116186724395528, 3.54048434342297253000106235990, 5.70406383875546182452588870599, 7.19649503812756783822299610539, 9.463742439339232452374340233725, 9.950068406572632759658882952468, 12.21530391491336415627324254040, 13.78454184469069281023438949660, 16.27731134020911919704747936826