L(s) = 1 | − 4.67e5·2-s − 3.48e9·3-s − 1.98e12·4-s − 2.77e14·5-s + 1.62e15·6-s + 3.80e17·7-s + 1.95e18·8-s + 1.21e19·9-s + 1.29e20·10-s + 1.78e20·11-s + 6.90e21·12-s + 2.06e22·13-s − 1.77e23·14-s + 9.66e23·15-s + 3.44e24·16-s − 2.32e25·17-s − 5.68e24·18-s + 2.05e26·19-s + 5.48e26·20-s − 1.32e27·21-s − 8.31e25·22-s − 4.66e27·23-s − 6.80e27·24-s + 3.12e28·25-s − 9.64e27·26-s − 4.23e28·27-s − 7.53e29·28-s + ⋯ |
L(s) = 1 | − 0.315·2-s − 0.577·3-s − 0.900·4-s − 1.29·5-s + 0.181·6-s + 1.80·7-s + 0.598·8-s + 0.333·9-s + 0.409·10-s + 0.0798·11-s + 0.520·12-s + 0.301·13-s − 0.567·14-s + 0.750·15-s + 0.712·16-s − 1.38·17-s − 0.105·18-s + 1.25·19-s + 1.17·20-s − 1.04·21-s − 0.0251·22-s − 0.566·23-s − 0.345·24-s + 0.688·25-s − 0.0948·26-s − 0.192·27-s − 1.62·28-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 | \( 1 + 4.67e5T + 2.19e12T^{2} \) |
| 5 | \( 1 + 2.77e14T + 4.54e28T^{2} \) |
| 7 | \( 1 - 3.80e17T + 4.45e34T^{2} \) |
| 11 | \( 1 - 1.78e20T + 4.97e42T^{2} \) |
| 13 | \( 1 - 2.06e22T + 4.69e45T^{2} \) |
| 17 | \( 1 + 2.32e25T + 2.80e50T^{2} \) |
| 19 | \( 1 - 2.05e26T + 2.68e52T^{2} \) |
| 23 | \( 1 + 4.66e27T + 6.77e55T^{2} \) |
| 29 | \( 1 - 1.57e30T + 9.08e59T^{2} \) |
| 31 | \( 1 + 4.51e30T + 1.39e61T^{2} \) |
| 37 | \( 1 + 1.32e32T + 1.97e64T^{2} \) |
| 41 | \( 1 + 3.66e32T + 1.33e66T^{2} \) |
| 43 | \( 1 + 6.46e32T + 9.38e66T^{2} \) |
| 47 | \( 1 - 2.68e34T + 3.59e68T^{2} \) |
| 53 | \( 1 + 1.90e35T + 4.95e70T^{2} \) |
| 59 | \( 1 + 3.62e36T + 4.02e72T^{2} \) |
| 61 | \( 1 - 2.16e36T + 1.57e73T^{2} \) |
| 67 | \( 1 - 3.15e36T + 7.39e74T^{2} \) |
| 71 | \( 1 + 4.01e37T + 7.97e75T^{2} \) |
| 73 | \( 1 - 7.75e37T + 2.49e76T^{2} \) |
| 79 | \( 1 + 8.36e38T + 6.34e77T^{2} \) |
| 83 | \( 1 + 1.97e39T + 4.81e78T^{2} \) |
| 89 | \( 1 - 3.68e39T + 8.41e79T^{2} \) |
| 97 | \( 1 - 4.15e40T + 2.86e81T^{2} \) |
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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
−15.66372150565877819617480047559, −14.03274344861509385420085579411, −11.95486943206766288562081448307, −10.86642578261287351713840637660, −8.638567083630994171787083270292, −7.57016448783397434689341346615, −5.01867390629877425993820040050, −4.09730662598594763577981892957, −1.30839260264355861021295564827, 0,
1.30839260264355861021295564827, 4.09730662598594763577981892957, 5.01867390629877425993820040050, 7.57016448783397434689341346615, 8.638567083630994171787083270292, 10.86642578261287351713840637660, 11.95486943206766288562081448307, 14.03274344861509385420085579411, 15.66372150565877819617480047559