| L(s) = 1 | − 3.96e4·2-s − 4.78e6·3-s + 1.03e9·4-s + 8.07e9·5-s + 1.89e11·6-s − 6.78e11·7-s − 1.96e13·8-s + 2.28e13·9-s − 3.19e14·10-s + 1.86e15·11-s − 4.94e15·12-s + 1.25e16·13-s + 2.68e16·14-s − 3.86e16·15-s + 2.24e17·16-s + 7.42e17·17-s − 9.06e17·18-s − 2.34e18·19-s + 8.34e18·20-s + 3.24e18·21-s − 7.39e19·22-s − 7.79e19·23-s + 9.39e19·24-s − 1.21e20·25-s − 4.96e20·26-s − 1.09e20·27-s − 7.00e20·28-s + ⋯ |
| L(s) = 1 | − 1.70·2-s − 0.577·3-s + 1.92·4-s + 0.591·5-s + 0.987·6-s − 0.377·7-s − 1.57·8-s + 0.333·9-s − 1.01·10-s + 1.48·11-s − 1.11·12-s + 0.882·13-s + 0.646·14-s − 0.341·15-s + 0.777·16-s + 1.06·17-s − 0.569·18-s − 0.672·19-s + 1.13·20-s + 0.218·21-s − 2.53·22-s − 1.40·23-s + 0.911·24-s − 0.649·25-s − 1.50·26-s − 0.192·27-s − 0.727·28-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 | \( 1 + 3.96e4T + 5.36e8T^{2} \) |
| 5 | \( 1 - 8.07e9T + 1.86e20T^{2} \) |
| 11 | \( 1 - 1.86e15T + 1.58e30T^{2} \) |
| 13 | \( 1 - 1.25e16T + 2.01e32T^{2} \) |
| 17 | \( 1 - 7.42e17T + 4.81e35T^{2} \) |
| 19 | \( 1 + 2.34e18T + 1.21e37T^{2} \) |
| 23 | \( 1 + 7.79e19T + 3.09e39T^{2} \) |
| 29 | \( 1 + 1.13e21T + 2.56e42T^{2} \) |
| 31 | \( 1 - 6.91e20T + 1.77e43T^{2} \) |
| 37 | \( 1 + 2.02e22T + 3.00e45T^{2} \) |
| 41 | \( 1 + 3.30e23T + 5.89e46T^{2} \) |
| 43 | \( 1 - 5.53e23T + 2.34e47T^{2} \) |
| 47 | \( 1 - 7.39e22T + 3.09e48T^{2} \) |
| 53 | \( 1 + 1.54e25T + 1.00e50T^{2} \) |
| 59 | \( 1 + 2.59e23T + 2.26e51T^{2} \) |
| 61 | \( 1 + 5.70e25T + 5.95e51T^{2} \) |
| 67 | \( 1 - 2.18e26T + 9.04e52T^{2} \) |
| 71 | \( 1 - 1.16e27T + 4.85e53T^{2} \) |
| 73 | \( 1 - 1.47e27T + 1.08e54T^{2} \) |
| 79 | \( 1 + 6.71e25T + 1.07e55T^{2} \) |
| 83 | \( 1 + 1.06e28T + 4.50e55T^{2} \) |
| 89 | \( 1 + 3.22e27T + 3.40e56T^{2} \) |
| 97 | \( 1 - 5.14e28T + 4.13e57T^{2} \) |
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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
−11.17101151995226875995239057420, −10.02966196312702512491738983988, −9.302184370602119828428809537074, −8.073679648651416534539727564971, −6.65941997076533941192537064597, −5.93701389395633916271770612338, −3.76997659404660831412040385527, −1.90791469439755908267545116772, −1.16503569798432556351941192335, 0,
1.16503569798432556351941192335, 1.90791469439755908267545116772, 3.76997659404660831412040385527, 5.93701389395633916271770612338, 6.65941997076533941192537064597, 8.073679648651416534539727564971, 9.302184370602119828428809537074, 10.02966196312702512491738983988, 11.17101151995226875995239057420
