| L(s) = 1 | + 8.38e6·2-s + 1.82e11·3-s + 7.03e13·4-s − 1.56e16·5-s + 1.53e18·6-s − 2.73e19·7-s + 5.90e20·8-s + 6.67e21·9-s − 1.31e23·10-s + 2.62e24·11-s + 1.28e25·12-s + 1.62e26·13-s − 2.29e26·14-s − 2.86e27·15-s + 4.95e27·16-s − 1.16e29·17-s + 5.60e28·18-s − 1.10e30·19-s − 1.10e30·20-s − 4.99e30·21-s + 2.20e31·22-s + 1.97e31·23-s + 1.07e32·24-s − 4.64e32·25-s + 1.36e33·26-s − 3.63e33·27-s − 1.92e33·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.11·3-s + 0.5·4-s − 0.588·5-s + 0.790·6-s − 0.377·7-s + 0.353·8-s + 0.251·9-s − 0.416·10-s + 0.884·11-s + 0.559·12-s + 1.07·13-s − 0.267·14-s − 0.658·15-s + 0.250·16-s − 1.41·17-s + 0.177·18-s − 0.983·19-s − 0.294·20-s − 0.422·21-s + 0.625·22-s + 0.197·23-s + 0.395·24-s − 0.653·25-s + 0.762·26-s − 0.837·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(48-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+47/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(24)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{49}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 8.38e6T \) |
| 7 | \( 1 + 2.73e19T \) |
| good | 3 | \( 1 - 1.82e11T + 2.65e22T^{2} \) |
| 5 | \( 1 + 1.56e16T + 7.10e32T^{2} \) |
| 11 | \( 1 - 2.62e24T + 8.81e48T^{2} \) |
| 13 | \( 1 - 1.62e26T + 2.26e52T^{2} \) |
| 17 | \( 1 + 1.16e29T + 6.77e57T^{2} \) |
| 19 | \( 1 + 1.10e30T + 1.26e60T^{2} \) |
| 23 | \( 1 - 1.97e31T + 1.00e64T^{2} \) |
| 29 | \( 1 - 1.16e33T + 5.40e68T^{2} \) |
| 31 | \( 1 + 5.39e33T + 1.24e70T^{2} \) |
| 37 | \( 1 + 2.39e36T + 5.07e73T^{2} \) |
| 41 | \( 1 + 1.15e38T + 6.32e75T^{2} \) |
| 43 | \( 1 - 1.27e38T + 5.92e76T^{2} \) |
| 47 | \( 1 - 3.34e39T + 3.87e78T^{2} \) |
| 53 | \( 1 - 2.56e40T + 1.09e81T^{2} \) |
| 59 | \( 1 + 7.05e41T + 1.69e83T^{2} \) |
| 61 | \( 1 - 2.49e40T + 8.13e83T^{2} \) |
| 67 | \( 1 - 5.09e42T + 6.69e85T^{2} \) |
| 71 | \( 1 + 2.12e43T + 1.02e87T^{2} \) |
| 73 | \( 1 + 9.57e43T + 3.76e87T^{2} \) |
| 79 | \( 1 - 1.17e44T + 1.54e89T^{2} \) |
| 83 | \( 1 + 1.13e45T + 1.57e90T^{2} \) |
| 89 | \( 1 + 3.56e45T + 4.18e91T^{2} \) |
| 97 | \( 1 + 3.90e46T + 2.38e93T^{2} \) |
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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
−10.78275426179599226665133024015, −9.097043476596553292326737463706, −8.382489650775273389574330024701, −7.02909104437356320384664160938, −6.01794381502341210978440206965, −4.23819975897009603335295033011, −3.70710890964499521230627896492, −2.62815082126810079520483376114, −1.58422975732687941727031634677, 0,
1.58422975732687941727031634677, 2.62815082126810079520483376114, 3.70710890964499521230627896492, 4.23819975897009603335295033011, 6.01794381502341210978440206965, 7.02909104437356320384664160938, 8.382489650775273389574330024701, 9.097043476596553292326737463706, 10.78275426179599226665133024015
