| L(s) = 1 | + 36.9·2-s + 81·3-s + 856.·4-s + 528.·5-s + 2.99e3·6-s + 1.27e4·8-s + 6.56e3·9-s + 1.95e4·10-s + 3.29e4·11-s + 6.93e4·12-s + 2.31e4·13-s + 4.28e4·15-s + 3.27e4·16-s + 5.22e5·17-s + 2.42e5·18-s − 1.43e5·19-s + 4.52e5·20-s + 1.21e6·22-s + 4.09e5·23-s + 1.03e6·24-s − 1.67e6·25-s + 8.55e5·26-s + 5.31e5·27-s + 6.68e6·29-s + 1.58e6·30-s + 3.95e6·31-s − 5.30e6·32-s + ⋯ |
| L(s) = 1 | + 1.63·2-s + 0.577·3-s + 1.67·4-s + 0.378·5-s + 0.943·6-s + 1.09·8-s + 0.333·9-s + 0.618·10-s + 0.677·11-s + 0.965·12-s + 0.224·13-s + 0.218·15-s + 0.125·16-s + 1.51·17-s + 0.544·18-s − 0.252·19-s + 0.632·20-s + 1.10·22-s + 0.305·23-s + 0.634·24-s − 0.856·25-s + 0.367·26-s + 0.192·27-s + 1.75·29-s + 0.357·30-s + 0.769·31-s − 0.895·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(8.564077344\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.564077344\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 81T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 36.9T + 512T^{2} \) |
| 5 | \( 1 - 528.T + 1.95e6T^{2} \) |
| 11 | \( 1 - 3.29e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 2.31e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 5.22e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 1.43e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 4.09e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 6.68e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 3.95e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 3.67e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 3.13e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 4.12e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.54e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 1.36e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 9.77e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 5.22e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.39e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.42e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.50e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 5.81e7T + 1.19e17T^{2} \) |
| 83 | \( 1 - 6.33e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 8.82e7T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.11e9T + 7.60e17T^{2} \) |
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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
−11.85446470793479817393941439451, −10.49563612424010242996233729107, −9.359156681293330587247450815644, −8.007629897677348817573568558488, −6.69595962567220298893971836473, −5.80586470575515054043368401822, −4.62534330521071734580354686658, −3.58525413352106493240375674270, −2.64553913559362646645755984664, −1.30531190304353910516861387159,
1.30531190304353910516861387159, 2.64553913559362646645755984664, 3.58525413352106493240375674270, 4.62534330521071734580354686658, 5.80586470575515054043368401822, 6.69595962567220298893971836473, 8.007629897677348817573568558488, 9.359156681293330587247450815644, 10.49563612424010242996233729107, 11.85446470793479817393941439451
