| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s + 15.3·5-s − 6·6-s − 5.31·7-s + 8·8-s + 9·9-s + 30.6·10-s − 21.2·11-s − 12·12-s + 87.2·13-s − 10.6·14-s − 45.9·15-s + 16·16-s + 18.0·17-s + 18·18-s + 47.9·19-s + 61.2·20-s + 15.9·21-s − 42.5·22-s + 23·23-s − 24·24-s + 109.·25-s + 174.·26-s − 27·27-s − 21.2·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.36·5-s − 0.408·6-s − 0.286·7-s + 0.353·8-s + 0.333·9-s + 0.968·10-s − 0.582·11-s − 0.288·12-s + 1.86·13-s − 0.202·14-s − 0.790·15-s + 0.250·16-s + 0.257·17-s + 0.235·18-s + 0.578·19-s + 0.684·20-s + 0.165·21-s − 0.411·22-s + 0.208·23-s − 0.204·24-s + 0.876·25-s + 1.31·26-s − 0.192·27-s − 0.143·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.618716729\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.618716729\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 + 3T \) |
| 23 | \( 1 - 23T \) |
| good | 5 | \( 1 - 15.3T + 125T^{2} \) |
| 7 | \( 1 + 5.31T + 343T^{2} \) |
| 11 | \( 1 + 21.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 87.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 18.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 47.9T + 6.85e3T^{2} \) |
| 29 | \( 1 - 179.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 117.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 410.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 205.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 149.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 299.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 295.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 737.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 550.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 527.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 297.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 714.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 834.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 795.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 46.9T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.03e3T + 9.12e5T^{2} \) |
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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
−13.06297512713794133088165406303, −11.77132133178420240247562408527, −10.68506598721816554621254007694, −9.938604545736183278032629847190, −8.546084886113601534665212586362, −6.80208346104643020606869581440, −5.94226890369352567012204560882, −5.12393301609889072256066311336, −3.33485659241343596257120943457, −1.55272768742799469341643332450,
1.55272768742799469341643332450, 3.33485659241343596257120943457, 5.12393301609889072256066311336, 5.94226890369352567012204560882, 6.80208346104643020606869581440, 8.546084886113601534665212586362, 9.938604545736183278032629847190, 10.68506598721816554621254007694, 11.77132133178420240247562408527, 13.06297512713794133088165406303
