L(s) = 1 | − 3.68·2-s + 9.05·3-s + 5.57·4-s − 7.08·5-s − 33.3·6-s − 28.1·7-s + 8.92·8-s + 55.0·9-s + 26.1·10-s + 15.3·11-s + 50.5·12-s + 2.51·13-s + 103.·14-s − 64.2·15-s − 77.5·16-s − 202.·18-s + 14.3·19-s − 39.5·20-s − 255.·21-s − 56.3·22-s − 180.·23-s + 80.8·24-s − 74.7·25-s − 9.26·26-s + 254.·27-s − 157.·28-s − 41.2·29-s + ⋯ |
L(s) = 1 | − 1.30·2-s + 1.74·3-s + 0.697·4-s − 0.634·5-s − 2.27·6-s − 1.52·7-s + 0.394·8-s + 2.03·9-s + 0.826·10-s + 0.419·11-s + 1.21·12-s + 0.0536·13-s + 1.98·14-s − 1.10·15-s − 1.21·16-s − 2.65·18-s + 0.173·19-s − 0.442·20-s − 2.65·21-s − 0.546·22-s − 1.63·23-s + 0.687·24-s − 0.597·25-s − 0.0699·26-s + 1.81·27-s − 1.06·28-s − 0.264·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 \) |
good | 2 | \( 1 + 3.68T + 8T^{2} \) |
| 3 | \( 1 - 9.05T + 27T^{2} \) |
| 5 | \( 1 + 7.08T + 125T^{2} \) |
| 7 | \( 1 + 28.1T + 343T^{2} \) |
| 11 | \( 1 - 15.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 2.51T + 2.19e3T^{2} \) |
| 19 | \( 1 - 14.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 180.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 41.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 155.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 225.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 234.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 321.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 326.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 57.1T + 1.48e5T^{2} \) |
| 59 | \( 1 + 241.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 460.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 392.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 615.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 697.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 991.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 98.9T + 5.71e5T^{2} \) |
| 89 | \( 1 - 698.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.42e3T + 9.12e5T^{2} \) |
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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
−10.23638744196176783567292932546, −9.656627519564296421576201736731, −9.061247669979385287216420904327, −8.144556316714523104268523669334, −7.52949493756290906091143460718, −6.49848290981917626391094187866, −4.08812214672394281205152864829, −3.26112264527392909397665587489, −1.85954986848823420319817747141, 0,
1.85954986848823420319817747141, 3.26112264527392909397665587489, 4.08812214672394281205152864829, 6.49848290981917626391094187866, 7.52949493756290906091143460718, 8.144556316714523104268523669334, 9.061247669979385287216420904327, 9.656627519564296421576201736731, 10.23638744196176783567292932546