| L(s) = 1 | − 1.15·2-s + 3·3-s − 6.66·4-s − 7.99·5-s − 3.46·6-s + 12.3·7-s + 16.9·8-s + 9·9-s + 9.23·10-s + 11·11-s − 19.9·12-s − 60.0·13-s − 14.2·14-s − 23.9·15-s + 33.7·16-s + 44.4·17-s − 10.3·18-s + 56.9·19-s + 53.3·20-s + 37.1·21-s − 12.7·22-s + 138.·23-s + 50.8·24-s − 61.0·25-s + 69.3·26-s + 27·27-s − 82.5·28-s + ⋯ |
| L(s) = 1 | − 0.408·2-s + 0.577·3-s − 0.833·4-s − 0.715·5-s − 0.235·6-s + 0.668·7-s + 0.748·8-s + 0.333·9-s + 0.291·10-s + 0.301·11-s − 0.481·12-s − 1.28·13-s − 0.272·14-s − 0.412·15-s + 0.527·16-s + 0.634·17-s − 0.136·18-s + 0.687·19-s + 0.595·20-s + 0.385·21-s − 0.123·22-s + 1.25·23-s + 0.432·24-s − 0.488·25-s + 0.523·26-s + 0.192·27-s − 0.556·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.501535556\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.501535556\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 11 | \( 1 - 11T \) |
| 61 | \( 1 + 61T \) |
| good | 2 | \( 1 + 1.15T + 8T^{2} \) |
| 5 | \( 1 + 7.99T + 125T^{2} \) |
| 7 | \( 1 - 12.3T + 343T^{2} \) |
| 13 | \( 1 + 60.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 44.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 56.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 138.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 41.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 222.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 119.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 355.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 419.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 606.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 53.3T + 1.48e5T^{2} \) |
| 59 | \( 1 - 103.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 313.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 716.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 660.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 768.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 556.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 265.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.34e3T + 9.12e5T^{2} \) |
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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
−8.739699260853981523411578446637, −7.941180567405143689475245112431, −7.67264920566966258193943829034, −6.74774629251088051620639881782, −5.20124015145048377358488233037, −4.79942922914196461198637018641, −3.84494397236245700369435346599, −2.99280229125662337128576344174, −1.64504206866953532183216705731, −0.61298603341766287479278836780,
0.61298603341766287479278836780, 1.64504206866953532183216705731, 2.99280229125662337128576344174, 3.84494397236245700369435346599, 4.79942922914196461198637018641, 5.20124015145048377358488233037, 6.74774629251088051620639881782, 7.67264920566966258193943829034, 7.941180567405143689475245112431, 8.739699260853981523411578446637
