| L(s) = 1 | + 4·2-s + 17.4·3-s + 16·4-s + 69.9·6-s − 132.·7-s + 64·8-s + 62.5·9-s + 311.·11-s + 279.·12-s − 901.·13-s − 531.·14-s + 256·16-s + 157.·17-s + 250.·18-s + 361·19-s − 2.32e3·21-s + 1.24e3·22-s + 2.52e3·23-s + 1.11e3·24-s − 3.60e3·26-s − 3.15e3·27-s − 2.12e3·28-s + 4.73e3·29-s − 6.58e3·31-s + 1.02e3·32-s + 5.44e3·33-s + 631.·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.12·3-s + 0.5·4-s + 0.792·6-s − 1.02·7-s + 0.353·8-s + 0.257·9-s + 0.776·11-s + 0.560·12-s − 1.47·13-s − 0.724·14-s + 0.250·16-s + 0.132·17-s + 0.182·18-s + 0.229·19-s − 1.14·21-s + 0.548·22-s + 0.994·23-s + 0.396·24-s − 1.04·26-s − 0.832·27-s − 0.512·28-s + 1.04·29-s − 1.23·31-s + 0.176·32-s + 0.870·33-s + 0.0936·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4T \) |
| 5 | \( 1 \) |
| 19 | \( 1 - 361T \) |
| good | 3 | \( 1 - 17.4T + 243T^{2} \) |
| 7 | \( 1 + 132.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 311.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 901.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 157.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 2.52e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.73e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.58e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 8.50e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.97e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.09e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.50e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.16e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.06e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 6.15e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.27e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.53e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.85e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.10e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.10e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.70e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.39e5T + 8.58e9T^{2} \) |
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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
−9.129434144609116193537200684772, −7.926825121170063782183372099451, −7.14630541703623830423773226349, −6.41566966873357572970608230544, −5.29162062857896153583668196511, −4.28163452685650426557895937720, −3.20689990266173227905758790300, −2.83144276114041174353568395192, −1.62859656729032328239270440338, 0,
1.62859656729032328239270440338, 2.83144276114041174353568395192, 3.20689990266173227905758790300, 4.28163452685650426557895937720, 5.29162062857896153583668196511, 6.41566966873357572970608230544, 7.14630541703623830423773226349, 7.926825121170063782183372099451, 9.129434144609116193537200684772
