| L(s) = 1 | + 162.·2-s + 729·3-s + 1.83e4·4-s + 2.77e4·5-s + 1.18e5·6-s − 5.95e4·7-s + 1.65e6·8-s + 5.31e5·9-s + 4.52e6·10-s − 2.68e6·11-s + 1.33e7·12-s − 3.02e6·13-s − 9.70e6·14-s + 2.02e7·15-s + 1.18e8·16-s + 1.27e8·17-s + 8.65e7·18-s + 3.71e8·19-s + 5.08e8·20-s − 4.34e7·21-s − 4.36e8·22-s − 8.99e8·23-s + 1.20e9·24-s − 4.49e8·25-s − 4.93e8·26-s + 3.87e8·27-s − 1.09e9·28-s + ⋯ |
| L(s) = 1 | + 1.79·2-s + 0.577·3-s + 2.23·4-s + 0.794·5-s + 1.03·6-s − 0.191·7-s + 2.22·8-s + 0.333·9-s + 1.42·10-s − 0.456·11-s + 1.29·12-s − 0.174·13-s − 0.344·14-s + 0.458·15-s + 1.76·16-s + 1.27·17-s + 0.599·18-s + 1.80·19-s + 1.77·20-s − 0.110·21-s − 0.821·22-s − 1.26·23-s + 1.28·24-s − 0.368·25-s − 0.313·26-s + 0.192·27-s − 0.428·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(12.27874079\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.27874079\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 729T \) |
| 59 | \( 1 + 4.21e10T \) |
| good | 2 | \( 1 - 162.T + 8.19e3T^{2} \) |
| 5 | \( 1 - 2.77e4T + 1.22e9T^{2} \) |
| 7 | \( 1 + 5.95e4T + 9.68e10T^{2} \) |
| 11 | \( 1 + 2.68e6T + 3.45e13T^{2} \) |
| 13 | \( 1 + 3.02e6T + 3.02e14T^{2} \) |
| 17 | \( 1 - 1.27e8T + 9.90e15T^{2} \) |
| 19 | \( 1 - 3.71e8T + 4.20e16T^{2} \) |
| 23 | \( 1 + 8.99e8T + 5.04e17T^{2} \) |
| 29 | \( 1 - 3.38e9T + 1.02e19T^{2} \) |
| 31 | \( 1 - 4.54e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + 1.05e10T + 2.43e20T^{2} \) |
| 41 | \( 1 + 1.90e10T + 9.25e20T^{2} \) |
| 43 | \( 1 - 6.35e10T + 1.71e21T^{2} \) |
| 47 | \( 1 - 9.70e10T + 5.46e21T^{2} \) |
| 53 | \( 1 - 2.65e11T + 2.60e22T^{2} \) |
| 61 | \( 1 - 1.69e11T + 1.61e23T^{2} \) |
| 67 | \( 1 - 4.94e11T + 5.48e23T^{2} \) |
| 71 | \( 1 - 2.21e10T + 1.16e24T^{2} \) |
| 73 | \( 1 - 4.50e11T + 1.67e24T^{2} \) |
| 79 | \( 1 + 1.95e12T + 4.66e24T^{2} \) |
| 83 | \( 1 + 3.51e12T + 8.87e24T^{2} \) |
| 89 | \( 1 - 1.52e12T + 2.19e25T^{2} \) |
| 97 | \( 1 - 6.38e12T + 6.73e25T^{2} \) |
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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
−10.35884860840724796602911243896, −9.671372214414533517424372558123, −7.996659318773873018612953633118, −7.03782108604176151422021411269, −5.83994681373315944118532286456, −5.28942808030976874057227073080, −4.04377698625612992075863577703, −3.07116404955206358086786443374, −2.35797562519475519258351487827, −1.18833937993312865135360979004,
1.18833937993312865135360979004, 2.35797562519475519258351487827, 3.07116404955206358086786443374, 4.04377698625612992075863577703, 5.28942808030976874057227073080, 5.83994681373315944118532286456, 7.03782108604176151422021411269, 7.996659318773873018612953633118, 9.671372214414533517424372558123, 10.35884860840724796602911243896
