| L(s) = 1 | + 4.70·2-s + 3·3-s + 14.1·4-s − 5·5-s + 14.1·6-s + 7·7-s + 28.7·8-s + 9·9-s − 23.5·10-s + 24.5·11-s + 42.3·12-s − 35.0·13-s + 32.9·14-s − 15·15-s + 22.1·16-s − 18.4·17-s + 42.3·18-s − 67.4·19-s − 70.5·20-s + 21·21-s + 115.·22-s − 145.·23-s + 86.1·24-s + 25·25-s − 164.·26-s + 27·27-s + 98.7·28-s + ⋯ |
| L(s) = 1 | + 1.66·2-s + 0.577·3-s + 1.76·4-s − 0.447·5-s + 0.959·6-s + 0.377·7-s + 1.26·8-s + 0.333·9-s − 0.743·10-s + 0.674·11-s + 1.01·12-s − 0.747·13-s + 0.628·14-s − 0.258·15-s + 0.345·16-s − 0.262·17-s + 0.554·18-s − 0.813·19-s − 0.788·20-s + 0.218·21-s + 1.12·22-s − 1.32·23-s + 0.732·24-s + 0.200·25-s − 1.24·26-s + 0.192·27-s + 0.666·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.188768699\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.188768699\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 5 | \( 1 + 5T \) |
| 7 | \( 1 - 7T \) |
| good | 2 | \( 1 - 4.70T + 8T^{2} \) |
| 11 | \( 1 - 24.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 35.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 18.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 67.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 145.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 214.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 88.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 162.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 337.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 122.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 354.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 676.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 501.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 708.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 907.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 430.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 41.3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 890.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.05e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.47e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 555.T + 9.12e5T^{2} \) |
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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
−13.45661982419435445974030247562, −12.31294012545683469886175642022, −11.74340904356697436312554036667, −10.38833133752295853137547898109, −8.762382598156180586820762710996, −7.41717170158120664380132296828, −6.24001980998692394717884292756, −4.71333953154907923884078271650, −3.82511888816381233528132119600, −2.32086255885988398607874075783,
2.32086255885988398607874075783, 3.82511888816381233528132119600, 4.71333953154907923884078271650, 6.24001980998692394717884292756, 7.41717170158120664380132296828, 8.762382598156180586820762710996, 10.38833133752295853137547898109, 11.74340904356697436312554036667, 12.31294012545683469886175642022, 13.45661982419435445974030247562
