| L(s) = 1 | + 3.52e4·2-s − 4.78e6·3-s + 7.03e8·4-s − 9.52e9·5-s − 1.68e11·6-s − 6.78e11·7-s + 5.86e12·8-s + 2.28e13·9-s − 3.35e14·10-s + 1.40e15·11-s − 3.36e15·12-s + 9.14e15·13-s − 2.38e16·14-s + 4.55e16·15-s − 1.71e17·16-s + 5.42e17·17-s + 8.05e17·18-s + 3.51e17·19-s − 6.70e18·20-s + 3.24e18·21-s + 4.95e19·22-s + 5.57e19·23-s − 2.80e19·24-s − 9.54e19·25-s + 3.22e20·26-s − 1.09e20·27-s − 4.77e20·28-s + ⋯ |
| L(s) = 1 | + 1.51·2-s − 0.577·3-s + 1.31·4-s − 0.698·5-s − 0.877·6-s − 0.377·7-s + 0.471·8-s + 0.333·9-s − 1.06·10-s + 1.11·11-s − 0.756·12-s + 0.644·13-s − 0.574·14-s + 0.403·15-s − 0.593·16-s + 0.781·17-s + 0.506·18-s + 0.100·19-s − 0.914·20-s + 0.218·21-s + 1.69·22-s + 1.00·23-s − 0.272·24-s − 0.512·25-s + 0.979·26-s − 0.192·27-s − 0.495·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(15)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{31}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 4.78e6T \) |
| 7 | \( 1 + 6.78e11T \) |
| good | 2 | \( 1 - 3.52e4T + 5.36e8T^{2} \) |
| 5 | \( 1 + 9.52e9T + 1.86e20T^{2} \) |
| 11 | \( 1 - 1.40e15T + 1.58e30T^{2} \) |
| 13 | \( 1 - 9.14e15T + 2.01e32T^{2} \) |
| 17 | \( 1 - 5.42e17T + 4.81e35T^{2} \) |
| 19 | \( 1 - 3.51e17T + 1.21e37T^{2} \) |
| 23 | \( 1 - 5.57e19T + 3.09e39T^{2} \) |
| 29 | \( 1 + 7.67e20T + 2.56e42T^{2} \) |
| 31 | \( 1 + 6.48e21T + 1.77e43T^{2} \) |
| 37 | \( 1 + 1.03e23T + 3.00e45T^{2} \) |
| 41 | \( 1 - 2.60e23T + 5.89e46T^{2} \) |
| 43 | \( 1 + 7.85e23T + 2.34e47T^{2} \) |
| 47 | \( 1 - 1.77e24T + 3.09e48T^{2} \) |
| 53 | \( 1 + 8.34e24T + 1.00e50T^{2} \) |
| 59 | \( 1 + 3.56e25T + 2.26e51T^{2} \) |
| 61 | \( 1 + 1.05e26T + 5.95e51T^{2} \) |
| 67 | \( 1 + 7.05e25T + 9.04e52T^{2} \) |
| 71 | \( 1 - 2.69e26T + 4.85e53T^{2} \) |
| 73 | \( 1 - 5.56e26T + 1.08e54T^{2} \) |
| 79 | \( 1 + 7.90e26T + 1.07e55T^{2} \) |
| 83 | \( 1 + 1.26e27T + 4.50e55T^{2} \) |
| 89 | \( 1 + 2.81e27T + 3.40e56T^{2} \) |
| 97 | \( 1 - 4.86e28T + 4.13e57T^{2} \) |
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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
−11.87510839524283972557101005324, −10.94930859082677716238938588550, −9.133768228172422239258880109063, −7.26410695301164287044840348805, −6.22323233604688638613074570325, −5.20074959458881697814387369708, −3.93367138312897892221419372135, −3.34153021988595031205995732488, −1.50584942277206848960041647493, 0,
1.50584942277206848960041647493, 3.34153021988595031205995732488, 3.93367138312897892221419372135, 5.20074959458881697814387369708, 6.22323233604688638613074570325, 7.26410695301164287044840348805, 9.133768228172422239258880109063, 10.94930859082677716238938588550, 11.87510839524283972557101005324
