| L(s) = 1 | + 3.73·2-s − 10.1·3-s + 5.92·4-s − 5·5-s − 38.0·6-s − 24.9·7-s − 7.73·8-s + 76.9·9-s − 18.6·10-s + 0.196·11-s − 60.4·12-s + 13·13-s − 93.0·14-s + 50.9·15-s − 76.2·16-s − 55.4·17-s + 287.·18-s + 22.4·19-s − 29.6·20-s + 254.·21-s + 0.732·22-s + 22.1·23-s + 78.8·24-s + 25·25-s + 48.5·26-s − 509.·27-s − 147.·28-s + ⋯ |
| L(s) = 1 | + 1.31·2-s − 1.96·3-s + 0.741·4-s − 0.447·5-s − 2.58·6-s − 1.34·7-s − 0.341·8-s + 2.85·9-s − 0.590·10-s + 0.00537·11-s − 1.45·12-s + 0.277·13-s − 1.77·14-s + 0.877·15-s − 1.19·16-s − 0.791·17-s + 3.76·18-s + 0.271·19-s − 0.331·20-s + 2.64·21-s + 0.00709·22-s + 0.201·23-s + 0.670·24-s + 0.200·25-s + 0.365·26-s − 3.63·27-s − 0.997·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 13 | \( 1 - 13T \) |
| good | 2 | \( 1 - 3.73T + 8T^{2} \) |
| 3 | \( 1 + 10.1T + 27T^{2} \) |
| 7 | \( 1 + 24.9T + 343T^{2} \) |
| 11 | \( 1 - 0.196T + 1.33e3T^{2} \) |
| 17 | \( 1 + 55.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 22.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 22.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 12.9T + 2.43e4T^{2} \) |
| 31 | \( 1 + 262.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 359.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 305.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 303.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 84.5T + 1.03e5T^{2} \) |
| 53 | \( 1 + 362.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 536.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 289.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 650.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 123.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 5.20T + 3.89e5T^{2} \) |
| 79 | \( 1 - 137.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 406.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 584.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.28e3T + 9.12e5T^{2} \) |
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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
−13.25068017122796700322942660805, −12.78308524222723564798483392504, −11.81317234699524588474001775896, −10.98399078720945423091435141230, −9.600831854272077638197250321282, −6.89648444501404754927551482652, −6.16335755994368370322516930220, −5.02784219651273827019372306164, −3.76856316024724878118723134822, 0,
3.76856316024724878118723134822, 5.02784219651273827019372306164, 6.16335755994368370322516930220, 6.89648444501404754927551482652, 9.600831854272077638197250321282, 10.98399078720945423091435141230, 11.81317234699524588474001775896, 12.78308524222723564798483392504, 13.25068017122796700322942660805
