| L(s) = 1 | − 3.73·2-s + 3·3-s + 5.95·4-s − 3.90·5-s − 11.2·6-s + 36.4·7-s + 7.64·8-s + 9·9-s + 14.5·10-s + 19.1·11-s + 17.8·12-s + 13·13-s − 136.·14-s − 11.7·15-s − 76.1·16-s − 83.8·17-s − 33.6·18-s + 46.8·19-s − 23.2·20-s + 109.·21-s − 71.6·22-s + 103.·23-s + 22.9·24-s − 109.·25-s − 48.5·26-s + 27·27-s + 216.·28-s + ⋯ |
| L(s) = 1 | − 1.32·2-s + 0.577·3-s + 0.744·4-s − 0.349·5-s − 0.762·6-s + 1.96·7-s + 0.337·8-s + 0.333·9-s + 0.461·10-s + 0.526·11-s + 0.429·12-s + 0.277·13-s − 2.59·14-s − 0.201·15-s − 1.19·16-s − 1.19·17-s − 0.440·18-s + 0.565·19-s − 0.260·20-s + 1.13·21-s − 0.694·22-s + 0.941·23-s + 0.195·24-s − 0.877·25-s − 0.366·26-s + 0.192·27-s + 1.46·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9014607473\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9014607473\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 13 | \( 1 - 13T \) |
| good | 2 | \( 1 + 3.73T + 8T^{2} \) |
| 5 | \( 1 + 3.90T + 125T^{2} \) |
| 7 | \( 1 - 36.4T + 343T^{2} \) |
| 11 | \( 1 - 19.1T + 1.33e3T^{2} \) |
| 17 | \( 1 + 83.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 46.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 103.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 108.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 147.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 160.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 231.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 340.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 119.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 732.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 229.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 108.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 10.3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 869.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.09e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 140.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 159.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.06e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 858.T + 9.12e5T^{2} \) |
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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
−15.86063762518602394068681098741, −14.70689240075525421131845151262, −13.63650315395773307900783175105, −11.60312994981776298364481036927, −10.76958401161466343610986832455, −9.125313057728331244327295481628, −8.305794915497233695757295811161, −7.32085853754735666088643676370, −4.56649153778551490017582005643, −1.61182073826344544636161528361,
1.61182073826344544636161528361, 4.56649153778551490017582005643, 7.32085853754735666088643676370, 8.305794915497233695757295811161, 9.125313057728331244327295481628, 10.76958401161466343610986832455, 11.60312994981776298364481036927, 13.63650315395773307900783175105, 14.70689240075525421131845151262, 15.86063762518602394068681098741
