| L(s) = 1 | + 4.01·2-s + 81·3-s − 495.·4-s + 2.14e3·5-s + 325.·6-s − 4.04e3·8-s + 6.56e3·9-s + 8.61e3·10-s + 1.17e4·11-s − 4.01e4·12-s + 2.55e4·13-s + 1.73e5·15-s + 2.37e5·16-s − 5.15e5·17-s + 2.63e4·18-s + 9.14e5·19-s − 1.06e6·20-s + 4.73e4·22-s + 1.27e6·23-s − 3.27e5·24-s + 2.64e6·25-s + 1.02e5·26-s + 5.31e5·27-s + 6.27e5·29-s + 6.98e5·30-s − 8.40e6·31-s + 3.02e6·32-s + ⋯ |
| L(s) = 1 | + 0.177·2-s + 0.577·3-s − 0.968·4-s + 1.53·5-s + 0.102·6-s − 0.349·8-s + 0.333·9-s + 0.272·10-s + 0.242·11-s − 0.559·12-s + 0.248·13-s + 0.886·15-s + 0.906·16-s − 1.49·17-s + 0.0591·18-s + 1.61·19-s − 1.48·20-s + 0.0430·22-s + 0.946·23-s − 0.201·24-s + 1.35·25-s + 0.0441·26-s + 0.192·27-s + 0.164·29-s + 0.157·30-s − 1.63·31-s + 0.510·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(3.350076280\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.350076280\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 81T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 4.01T + 512T^{2} \) |
| 5 | \( 1 - 2.14e3T + 1.95e6T^{2} \) |
| 11 | \( 1 - 1.17e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 2.55e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 5.15e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 9.14e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.27e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 6.27e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + 8.40e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 4.47e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.59e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.20e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.77e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 1.00e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.64e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.22e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.19e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.90e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.18e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.73e7T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.87e7T + 1.86e17T^{2} \) |
| 89 | \( 1 + 2.34e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 9.57e8T + 7.60e17T^{2} \) |
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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
−11.24703278953385092920885046567, −9.908783748240421806992896917984, −9.319054614694692490158292663841, −8.603483315764204774894841332920, −7.04110766976876599220474955099, −5.77005178429734971201258724653, −4.84909681567215281793353012113, −3.46817008308963582902308102065, −2.17150040236109093309250596198, −0.938734204671283731632005305352,
0.938734204671283731632005305352, 2.17150040236109093309250596198, 3.46817008308963582902308102065, 4.84909681567215281793353012113, 5.77005178429734971201258724653, 7.04110766976876599220474955099, 8.603483315764204774894841332920, 9.319054614694692490158292663841, 9.908783748240421806992896917984, 11.24703278953385092920885046567
