| L(s) = 1 | + 0.832·2-s − 19.4·3-s − 31.3·4-s − 25·5-s − 16.2·6-s + 142.·7-s − 52.6·8-s + 136.·9-s − 20.8·10-s − 121·11-s + 610.·12-s + 1.03e3·13-s + 118.·14-s + 487.·15-s + 958.·16-s + 273.·17-s + 113.·18-s + 361·19-s + 782.·20-s − 2.77e3·21-s − 100.·22-s + 1.67e3·23-s + 1.02e3·24-s + 625·25-s + 860.·26-s + 2.07e3·27-s − 4.46e3·28-s + ⋯ |
| L(s) = 1 | + 0.147·2-s − 1.25·3-s − 0.978·4-s − 0.447·5-s − 0.183·6-s + 1.10·7-s − 0.291·8-s + 0.562·9-s − 0.0657·10-s − 0.301·11-s + 1.22·12-s + 1.69·13-s + 0.161·14-s + 0.559·15-s + 0.935·16-s + 0.229·17-s + 0.0827·18-s + 0.229·19-s + 0.437·20-s − 1.37·21-s − 0.0443·22-s + 0.660·23-s + 0.363·24-s + 0.200·25-s + 0.249·26-s + 0.546·27-s − 1.07·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.426395995\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.426395995\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 + 121T \) |
| 19 | \( 1 - 361T \) |
| good | 2 | \( 1 - 0.832T + 32T^{2} \) |
| 3 | \( 1 + 19.4T + 243T^{2} \) |
| 7 | \( 1 - 142.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 1.03e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 273.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 1.67e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.56e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.99e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 338.T + 6.93e7T^{2} \) |
| 41 | \( 1 + 4.41e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.95e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 9.83e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.34e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.91e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.01e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.18e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.03e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.27e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.62e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.82e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.08e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 5.43e4T + 8.58e9T^{2} \) |
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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
−8.983898077253160075326718042894, −8.421630401982005688950604991823, −7.63183984525685287171743674821, −6.36336297231593299860523042496, −5.65881642047701206428007041021, −4.85204368469403070340678772140, −4.29003208186803830526553376428, −3.13217181480878796179461370794, −1.23205184042833501733445061023, −0.66383557065253364591912331025,
0.66383557065253364591912331025, 1.23205184042833501733445061023, 3.13217181480878796179461370794, 4.29003208186803830526553376428, 4.85204368469403070340678772140, 5.65881642047701206428007041021, 6.36336297231593299860523042496, 7.63183984525685287171743674821, 8.421630401982005688950604991823, 8.983898077253160075326718042894
