| L(s) = 1 | + 5.40·2-s − 27·3-s − 98.7·4-s − 344.·5-s − 145.·6-s + 525.·7-s − 1.22e3·8-s + 729·9-s − 1.86e3·10-s − 6.69e3·11-s + 2.66e3·12-s + 3.86e3·13-s + 2.84e3·14-s + 9.29e3·15-s + 6.01e3·16-s + 4.99e3·17-s + 3.94e3·18-s + 5.23e4·19-s + 3.39e4·20-s − 1.41e4·21-s − 3.61e4·22-s − 1.21e4·23-s + 3.31e4·24-s + 4.03e4·25-s + 2.08e4·26-s − 1.96e4·27-s − 5.19e4·28-s + ⋯ |
| L(s) = 1 | + 0.477·2-s − 0.577·3-s − 0.771·4-s − 1.23·5-s − 0.275·6-s + 0.579·7-s − 0.846·8-s + 0.333·9-s − 0.588·10-s − 1.51·11-s + 0.445·12-s + 0.487·13-s + 0.276·14-s + 0.710·15-s + 0.367·16-s + 0.246·17-s + 0.159·18-s + 1.75·19-s + 0.950·20-s − 0.334·21-s − 0.724·22-s − 0.208·23-s + 0.488·24-s + 0.516·25-s + 0.233·26-s − 0.192·27-s − 0.447·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.9731853448\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9731853448\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 27T \) |
| 23 | \( 1 + 1.21e4T \) |
| good | 2 | \( 1 - 5.40T + 128T^{2} \) |
| 5 | \( 1 + 344.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 525.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 6.69e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 3.86e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 4.99e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 5.23e4T + 8.93e8T^{2} \) |
| 29 | \( 1 - 6.93e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 7.43e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.45e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.04e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 3.34e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 6.85e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 7.59e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 8.93e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.01e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.90e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.91e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.34e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 5.28e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 4.24e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 4.56e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.25e7T + 8.07e13T^{2} \) |
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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
−13.21993311049895632275948975666, −12.15338138208211698982643231284, −11.33537066203441731636149046079, −10.05049695255120469086228786156, −8.395285495124545419868730361250, −7.52467658044145302511622838259, −5.55224636844274548771773056582, −4.65807705374328775686013171566, −3.33198625383318544555578519098, −0.63888490757203741717678465888,
0.63888490757203741717678465888, 3.33198625383318544555578519098, 4.65807705374328775686013171566, 5.55224636844274548771773056582, 7.52467658044145302511622838259, 8.395285495124545419868730361250, 10.05049695255120469086228786156, 11.33537066203441731636149046079, 12.15338138208211698982643231284, 13.21993311049895632275948975666
