| L(s) = 1 | + 3.76e4·2-s − 4.78e6·3-s + 8.79e8·4-s + 1.83e10·5-s − 1.79e11·6-s − 6.78e11·7-s + 1.28e13·8-s + 2.28e13·9-s + 6.92e14·10-s − 7.53e14·11-s − 4.20e15·12-s − 1.44e16·13-s − 2.55e16·14-s − 8.79e16·15-s + 1.29e16·16-s − 8.99e17·17-s + 8.60e17·18-s + 1.24e18·19-s + 1.61e19·20-s + 3.24e18·21-s − 2.83e19·22-s − 3.80e19·23-s − 6.16e19·24-s + 1.52e20·25-s − 5.42e20·26-s − 1.09e20·27-s − 5.96e20·28-s + ⋯ |
| L(s) = 1 | + 1.62·2-s − 0.577·3-s + 1.63·4-s + 1.34·5-s − 0.937·6-s − 0.377·7-s + 1.03·8-s + 0.333·9-s + 2.18·10-s − 0.598·11-s − 0.945·12-s − 1.01·13-s − 0.613·14-s − 0.778·15-s + 0.0448·16-s − 1.29·17-s + 0.541·18-s + 0.358·19-s + 2.20·20-s + 0.218·21-s − 0.972·22-s − 0.684·23-s − 0.598·24-s + 0.817·25-s − 1.65·26-s − 0.192·27-s − 0.619·28-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 | \( 1 - 3.76e4T + 5.36e8T^{2} \) |
| 5 | \( 1 - 1.83e10T + 1.86e20T^{2} \) |
| 11 | \( 1 + 7.53e14T + 1.58e30T^{2} \) |
| 13 | \( 1 + 1.44e16T + 2.01e32T^{2} \) |
| 17 | \( 1 + 8.99e17T + 4.81e35T^{2} \) |
| 19 | \( 1 - 1.24e18T + 1.21e37T^{2} \) |
| 23 | \( 1 + 3.80e19T + 3.09e39T^{2} \) |
| 29 | \( 1 + 1.55e21T + 2.56e42T^{2} \) |
| 31 | \( 1 + 1.79e20T + 1.77e43T^{2} \) |
| 37 | \( 1 - 4.46e22T + 3.00e45T^{2} \) |
| 41 | \( 1 + 3.43e23T + 5.89e46T^{2} \) |
| 43 | \( 1 - 5.56e23T + 2.34e47T^{2} \) |
| 47 | \( 1 - 1.17e24T + 3.09e48T^{2} \) |
| 53 | \( 1 - 4.55e24T + 1.00e50T^{2} \) |
| 59 | \( 1 - 1.57e25T + 2.26e51T^{2} \) |
| 61 | \( 1 + 2.37e25T + 5.95e51T^{2} \) |
| 67 | \( 1 + 2.63e26T + 9.04e52T^{2} \) |
| 71 | \( 1 + 1.36e27T + 4.85e53T^{2} \) |
| 73 | \( 1 - 3.24e26T + 1.08e54T^{2} \) |
| 79 | \( 1 - 4.00e27T + 1.07e55T^{2} \) |
| 83 | \( 1 + 1.21e28T + 4.50e55T^{2} \) |
| 89 | \( 1 + 4.32e27T + 3.40e56T^{2} \) |
| 97 | \( 1 - 1.14e29T + 4.13e57T^{2} \) |
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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
−11.93958163456348483435584072516, −10.61231678572576060406139785787, −9.431536129488326435597647103554, −7.10773795735439888536200634254, −6.04271123042262943166090772259, −5.35163604829453411211115689754, −4.30295820892292374045429548412, −2.72158499688290941430556499242, −1.92450534125781899366554624131, 0,
1.92450534125781899366554624131, 2.72158499688290941430556499242, 4.30295820892292374045429548412, 5.35163604829453411211115689754, 6.04271123042262943166090772259, 7.10773795735439888536200634254, 9.431536129488326435597647103554, 10.61231678572576060406139785787, 11.93958163456348483435584072516
