L(s) = 1 | − 237.·2-s − 1.41e4·3-s − 7.45e4·4-s + 1.22e6·5-s + 3.37e6·6-s + 1.05e7·7-s + 4.88e7·8-s + 7.20e7·9-s − 2.90e8·10-s + 9.82e8·11-s + 1.05e9·12-s + 5.20e9·13-s − 2.51e9·14-s − 1.73e10·15-s − 1.86e9·16-s − 2.48e10·17-s − 1.71e10·18-s + 7.35e10·19-s − 9.10e10·20-s − 1.49e11·21-s − 2.33e11·22-s − 4.63e11·23-s − 6.93e11·24-s + 7.29e11·25-s − 1.23e12·26-s + 8.09e11·27-s − 7.86e11·28-s + ⋯ |
L(s) = 1 | − 0.656·2-s − 1.24·3-s − 0.568·4-s + 1.39·5-s + 0.820·6-s + 0.692·7-s + 1.03·8-s + 0.558·9-s − 0.918·10-s + 1.38·11-s + 0.709·12-s + 1.77·13-s − 0.454·14-s − 1.74·15-s − 0.108·16-s − 0.863·17-s − 0.366·18-s + 0.993·19-s − 0.794·20-s − 0.863·21-s − 0.907·22-s − 1.23·23-s − 1.28·24-s + 0.956·25-s − 1.16·26-s + 0.551·27-s − 0.393·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(9)\) |
\(\approx\) |
\(1.405775071\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.405775071\) |
\(L(\frac{19}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 29 | \( 1 - 5.00e11T \) |
good | 2 | \( 1 + 237.T + 1.31e5T^{2} \) |
| 3 | \( 1 + 1.41e4T + 1.29e8T^{2} \) |
| 5 | \( 1 - 1.22e6T + 7.62e11T^{2} \) |
| 7 | \( 1 - 1.05e7T + 2.32e14T^{2} \) |
| 11 | \( 1 - 9.82e8T + 5.05e17T^{2} \) |
| 13 | \( 1 - 5.20e9T + 8.65e18T^{2} \) |
| 17 | \( 1 + 2.48e10T + 8.27e20T^{2} \) |
| 19 | \( 1 - 7.35e10T + 5.48e21T^{2} \) |
| 23 | \( 1 + 4.63e11T + 1.41e23T^{2} \) |
| 31 | \( 1 - 4.00e12T + 2.25e25T^{2} \) |
| 37 | \( 1 - 9.74e12T + 4.56e26T^{2} \) |
| 41 | \( 1 - 5.79e13T + 2.61e27T^{2} \) |
| 43 | \( 1 + 8.74e13T + 5.87e27T^{2} \) |
| 47 | \( 1 - 8.94e13T + 2.66e28T^{2} \) |
| 53 | \( 1 - 6.03e14T + 2.05e29T^{2} \) |
| 59 | \( 1 - 9.08e14T + 1.27e30T^{2} \) |
| 61 | \( 1 + 2.23e15T + 2.24e30T^{2} \) |
| 67 | \( 1 - 1.04e15T + 1.10e31T^{2} \) |
| 71 | \( 1 - 2.03e15T + 2.96e31T^{2} \) |
| 73 | \( 1 + 3.65e15T + 4.74e31T^{2} \) |
| 79 | \( 1 - 6.95e15T + 1.81e32T^{2} \) |
| 83 | \( 1 + 1.05e16T + 4.21e32T^{2} \) |
| 89 | \( 1 + 3.63e16T + 1.37e33T^{2} \) |
| 97 | \( 1 - 1.37e17T + 5.95e33T^{2} \) |
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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
−13.45509737135551415277588915454, −11.72456431746384482802319028891, −10.72339259313876738591249202581, −9.568205081757098329171617332079, −8.506739263630567185202379251386, −6.46327816078076640464976113180, −5.61238385387607476718305731074, −4.25758292376878231104200351477, −1.58748124921774700233115227453, −0.887972723781716850296972329589,
0.887972723781716850296972329589, 1.58748124921774700233115227453, 4.25758292376878231104200351477, 5.61238385387607476718305731074, 6.46327816078076640464976113180, 8.506739263630567185202379251386, 9.568205081757098329171617332079, 10.72339259313876738591249202581, 11.72456431746384482802319028891, 13.45509737135551415277588915454