| L(s) = 1 | + 8.38e6·2-s + 2.46e11·3-s + 7.03e13·4-s + 3.56e15·5-s + 2.07e18·6-s − 2.73e19·7-s + 5.90e20·8-s + 3.43e22·9-s + 2.98e22·10-s − 2.59e24·11-s + 1.73e25·12-s − 2.31e26·13-s − 2.29e26·14-s + 8.79e26·15-s + 4.95e27·16-s + 1.36e29·17-s + 2.88e29·18-s − 1.34e30·19-s + 2.50e29·20-s − 6.75e30·21-s − 2.17e31·22-s − 1.72e31·23-s + 1.45e32·24-s − 6.97e32·25-s − 1.94e33·26-s + 1.91e33·27-s − 1.92e33·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.51·3-s + 0.5·4-s + 0.133·5-s + 1.07·6-s − 0.377·7-s + 0.353·8-s + 1.29·9-s + 0.0945·10-s − 0.872·11-s + 0.756·12-s − 1.54·13-s − 0.267·14-s + 0.202·15-s + 0.250·16-s + 1.65·17-s + 0.913·18-s − 1.19·19-s + 0.0668·20-s − 0.572·21-s − 0.617·22-s − 0.172·23-s + 0.535·24-s − 0.982·25-s − 1.08·26-s + 0.440·27-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 | \( 1 - 2.46e11T + 2.65e22T^{2} \) |
| 5 | \( 1 - 3.56e15T + 7.10e32T^{2} \) |
| 11 | \( 1 + 2.59e24T + 8.81e48T^{2} \) |
| 13 | \( 1 + 2.31e26T + 2.26e52T^{2} \) |
| 17 | \( 1 - 1.36e29T + 6.77e57T^{2} \) |
| 19 | \( 1 + 1.34e30T + 1.26e60T^{2} \) |
| 23 | \( 1 + 1.72e31T + 1.00e64T^{2} \) |
| 29 | \( 1 + 2.52e34T + 5.40e68T^{2} \) |
| 31 | \( 1 - 1.43e35T + 1.24e70T^{2} \) |
| 37 | \( 1 + 1.22e37T + 5.07e73T^{2} \) |
| 41 | \( 1 - 1.49e38T + 6.32e75T^{2} \) |
| 43 | \( 1 - 2.17e37T + 5.92e76T^{2} \) |
| 47 | \( 1 + 9.47e38T + 3.87e78T^{2} \) |
| 53 | \( 1 + 5.77e40T + 1.09e81T^{2} \) |
| 59 | \( 1 - 3.13e41T + 1.69e83T^{2} \) |
| 61 | \( 1 + 1.36e42T + 8.13e83T^{2} \) |
| 67 | \( 1 + 5.70e42T + 6.69e85T^{2} \) |
| 71 | \( 1 + 1.20e43T + 1.02e87T^{2} \) |
| 73 | \( 1 + 7.60e43T + 3.76e87T^{2} \) |
| 79 | \( 1 - 3.72e44T + 1.54e89T^{2} \) |
| 83 | \( 1 + 9.76e44T + 1.57e90T^{2} \) |
| 89 | \( 1 + 4.02e45T + 4.18e91T^{2} \) |
| 97 | \( 1 - 6.97e46T + 2.38e93T^{2} \) |
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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
−10.31701988441894031099031040046, −9.489724533113955042948875396016, −8.028354799576649938525964501524, −7.37416207623089966436695391818, −5.79790965367412125856357373842, −4.52917342742684372164491627024, −3.35654617831339734567811133071, −2.61869457682723187364684334328, −1.79545122999772378078175043180, 0,
1.79545122999772378078175043180, 2.61869457682723187364684334328, 3.35654617831339734567811133071, 4.52917342742684372164491627024, 5.79790965367412125856357373842, 7.37416207623089966436695391818, 8.028354799576649938525964501524, 9.489724533113955042948875396016, 10.31701988441894031099031040046
