| L(s) = 1 | − 3.20·2-s − 7.51·3-s + 2.25·4-s − 15.1·5-s + 24.0·6-s + 8.04·7-s + 18.3·8-s + 29.4·9-s + 48.6·10-s − 16.9·12-s + 64.4·13-s − 25.7·14-s + 114.·15-s − 76.9·16-s − 17·17-s − 94.3·18-s − 147.·19-s − 34.2·20-s − 60.4·21-s + 134.·23-s − 138.·24-s + 106.·25-s − 206.·26-s − 18.4·27-s + 18.1·28-s − 149.·29-s − 365.·30-s + ⋯ |
| L(s) = 1 | − 1.13·2-s − 1.44·3-s + 0.281·4-s − 1.35·5-s + 1.63·6-s + 0.434·7-s + 0.813·8-s + 1.09·9-s + 1.53·10-s − 0.407·12-s + 1.37·13-s − 0.492·14-s + 1.96·15-s − 1.20·16-s − 0.242·17-s − 1.23·18-s − 1.78·19-s − 0.383·20-s − 0.628·21-s + 1.21·23-s − 1.17·24-s + 0.848·25-s − 1.55·26-s − 0.131·27-s + 0.122·28-s − 0.954·29-s − 2.22·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2057 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2057 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 17 | \( 1 + 17T \) |
| good | 2 | \( 1 + 3.20T + 8T^{2} \) |
| 3 | \( 1 + 7.51T + 27T^{2} \) |
| 5 | \( 1 + 15.1T + 125T^{2} \) |
| 7 | \( 1 - 8.04T + 343T^{2} \) |
| 13 | \( 1 - 64.4T + 2.19e3T^{2} \) |
| 19 | \( 1 + 147.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 134.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 149.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 30.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 169.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 68.6T + 6.89e4T^{2} \) |
| 43 | \( 1 - 46.4T + 7.95e4T^{2} \) |
| 47 | \( 1 + 58.3T + 1.03e5T^{2} \) |
| 53 | \( 1 + 422.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 674.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 393.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 852.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 682.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.02e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 830.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 138.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 277.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 402.T + 9.12e5T^{2} \) |
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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
−8.532006573184033544736379768482, −7.70472174354924735547690384993, −6.91570247664308679701097022444, −6.23886391249319704256757968690, −5.09478776045905088183240661618, −4.41615041279719740362168647747, −3.62703273512345568415504672540, −1.74232427422379909658993517261, −0.73275340847257844108611821268, 0,
0.73275340847257844108611821268, 1.74232427422379909658993517261, 3.62703273512345568415504672540, 4.41615041279719740362168647747, 5.09478776045905088183240661618, 6.23886391249319704256757968690, 6.91570247664308679701097022444, 7.70472174354924735547690384993, 8.532006573184033544736379768482
