| L(s) = 1 | − 123. i·2-s + 1.25e3·4-s − 4.86e4i·5-s + 1.10e6·7-s − 2.16e6i·8-s − 5.98e6·10-s + 2.78e7i·11-s + 8.92e7·13-s − 1.35e8i·14-s − 2.46e8·16-s + 1.83e8i·17-s + 1.31e9·19-s − 6.09e7i·20-s + 3.42e9·22-s + 9.96e8i·23-s + ⋯ |
| L(s) = 1 | − 0.961i·2-s + 0.0764·4-s − 0.622i·5-s + 1.34·7-s − 1.03i·8-s − 0.598·10-s + 1.42i·11-s + 1.42·13-s − 1.28i·14-s − 0.917·16-s + 0.446i·17-s + 1.47·19-s − 0.0476i·20-s + 1.37·22-s + 0.292i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{15}{2})\) |
\(\approx\) |
\(2.982647934\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.982647934\) |
| \(L(8)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + 123. iT - 1.63e4T^{2} \) |
| 5 | \( 1 + 4.86e4iT - 6.10e9T^{2} \) |
| 7 | \( 1 - 1.10e6T + 6.78e11T^{2} \) |
| 11 | \( 1 - 2.78e7iT - 3.79e14T^{2} \) |
| 13 | \( 1 - 8.92e7T + 3.93e15T^{2} \) |
| 17 | \( 1 - 1.83e8iT - 1.68e17T^{2} \) |
| 19 | \( 1 - 1.31e9T + 7.99e17T^{2} \) |
| 23 | \( 1 - 9.96e8iT - 1.15e19T^{2} \) |
| 29 | \( 1 - 4.05e8iT - 2.97e20T^{2} \) |
| 31 | \( 1 + 1.31e10T + 7.56e20T^{2} \) |
| 37 | \( 1 + 8.42e10T + 9.01e21T^{2} \) |
| 41 | \( 1 + 2.20e11iT - 3.79e22T^{2} \) |
| 43 | \( 1 + 2.37e11T + 7.38e22T^{2} \) |
| 47 | \( 1 + 8.39e11iT - 2.56e23T^{2} \) |
| 53 | \( 1 + 1.03e12iT - 1.37e24T^{2} \) |
| 59 | \( 1 - 3.62e12iT - 6.19e24T^{2} \) |
| 61 | \( 1 - 3.70e12T + 9.87e24T^{2} \) |
| 67 | \( 1 + 6.86e12T + 3.67e25T^{2} \) |
| 71 | \( 1 - 8.81e12iT - 8.27e25T^{2} \) |
| 73 | \( 1 - 1.29e13T + 1.22e26T^{2} \) |
| 79 | \( 1 + 1.51e13T + 3.68e26T^{2} \) |
| 83 | \( 1 + 3.80e13iT - 7.36e26T^{2} \) |
| 89 | \( 1 - 4.75e13iT - 1.95e27T^{2} \) |
| 97 | \( 1 - 6.50e13T + 6.52e27T^{2} \) |
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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
−13.46543265036270584044893990800, −12.23346692809295117424658226097, −11.36422474795303798730395549887, −10.18985892179214661065663668597, −8.708441828856820627623940785608, −7.20555337008872606633316183667, −5.16503495023528784552035737619, −3.77440219518191777119026560963, −1.87626653461201608758496140538, −1.14638841442017867707447375285,
1.28176801836597963345768035809, 3.12736131851130620561858812461, 5.19839700243101296122596989087, 6.34587462384121071781562762854, 7.72752781270692661246006518383, 8.660576882917516315506671340619, 10.98141797340259129990110903167, 11.42919617763991436745559380993, 13.80584960551579192097333195813, 14.41120885561103364061740275248
