L(s) = 1 | + 1.51e6·2-s + 1.04e10·3-s − 6.49e12·4-s + 1.66e15·5-s + 1.58e16·6-s − 2.14e18·7-s − 2.31e19·8-s + 1.09e20·9-s + 2.52e21·10-s + 1.22e22·11-s − 6.79e22·12-s − 1.24e24·13-s − 3.25e24·14-s + 1.74e25·15-s + 2.20e25·16-s − 3.24e26·17-s + 1.65e26·18-s + 1.94e27·19-s − 1.08e28·20-s − 2.24e28·21-s + 1.86e28·22-s − 1.30e28·23-s − 2.42e29·24-s + 1.64e30·25-s − 1.88e30·26-s + 1.14e30·27-s + 1.39e31·28-s + ⋯ |
L(s) = 1 | + 0.511·2-s + 0.577·3-s − 0.738·4-s + 1.56·5-s + 0.295·6-s − 1.45·7-s − 0.888·8-s + 0.333·9-s + 0.799·10-s + 0.500·11-s − 0.426·12-s − 1.39·13-s − 0.742·14-s + 0.903·15-s + 0.284·16-s − 1.13·17-s + 0.170·18-s + 0.625·19-s − 1.15·20-s − 0.838·21-s + 0.255·22-s − 0.0690·23-s − 0.513·24-s + 1.44·25-s − 0.713·26-s + 0.192·27-s + 1.07·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(44-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+43/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(22)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{45}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 1.04e10T \) |
good | 2 | \( 1 - 1.51e6T + 8.79e12T^{2} \) |
| 5 | \( 1 - 1.66e15T + 1.13e30T^{2} \) |
| 7 | \( 1 + 2.14e18T + 2.18e36T^{2} \) |
| 11 | \( 1 - 1.22e22T + 6.02e44T^{2} \) |
| 13 | \( 1 + 1.24e24T + 7.93e47T^{2} \) |
| 17 | \( 1 + 3.24e26T + 8.11e52T^{2} \) |
| 19 | \( 1 - 1.94e27T + 9.69e54T^{2} \) |
| 23 | \( 1 + 1.30e28T + 3.58e58T^{2} \) |
| 29 | \( 1 + 3.97e31T + 7.64e62T^{2} \) |
| 31 | \( 1 + 2.09e32T + 1.34e64T^{2} \) |
| 37 | \( 1 - 2.50e33T + 2.70e67T^{2} \) |
| 41 | \( 1 + 4.85e34T + 2.23e69T^{2} \) |
| 43 | \( 1 + 1.71e35T + 1.73e70T^{2} \) |
| 47 | \( 1 + 5.66e35T + 7.94e71T^{2} \) |
| 53 | \( 1 + 3.68e36T + 1.39e74T^{2} \) |
| 59 | \( 1 + 2.83e37T + 1.40e76T^{2} \) |
| 61 | \( 1 - 1.88e38T + 5.87e76T^{2} \) |
| 67 | \( 1 - 4.53e37T + 3.32e78T^{2} \) |
| 71 | \( 1 - 9.40e39T + 4.01e79T^{2} \) |
| 73 | \( 1 - 1.85e40T + 1.32e80T^{2} \) |
| 79 | \( 1 - 2.01e40T + 3.96e81T^{2} \) |
| 83 | \( 1 - 5.04e40T + 3.31e82T^{2} \) |
| 89 | \( 1 - 2.49e41T + 6.66e83T^{2} \) |
| 97 | \( 1 - 5.50e42T + 2.69e85T^{2} \) |
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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
−14.76106598841272696882443536687, −13.53244704435461337073574164602, −12.77781380583756211841277665729, −9.702647959702834978861160925650, −9.296499235348970975407832377435, −6.65277210365325693052747921469, −5.25975086481495194044518627305, −3.46990581254968739945337984940, −2.12413853973690171786007794616, 0,
2.12413853973690171786007794616, 3.46990581254968739945337984940, 5.25975086481495194044518627305, 6.65277210365325693052747921469, 9.296499235348970975407832377435, 9.702647959702834978861160925650, 12.77781380583756211841277665729, 13.53244704435461337073574164602, 14.76106598841272696882443536687