| L(s) = 1 | + 1.37·2-s − 1.01·3-s − 0.0996·4-s − 5-s − 1.39·6-s + 4.61·7-s − 2.89·8-s − 1.97·9-s − 1.37·10-s + 2.87·11-s + 0.100·12-s − 0.468·13-s + 6.36·14-s + 1.01·15-s − 3.79·16-s + 0.617·17-s − 2.72·18-s + 0.641·19-s + 0.0996·20-s − 4.66·21-s + 3.96·22-s − 3.06·23-s + 2.92·24-s + 25-s − 0.646·26-s + 5.03·27-s − 0.460·28-s + ⋯ |
| L(s) = 1 | + 0.974·2-s − 0.583·3-s − 0.0498·4-s − 0.447·5-s − 0.569·6-s + 1.74·7-s − 1.02·8-s − 0.659·9-s − 0.435·10-s + 0.867·11-s + 0.0290·12-s − 0.130·13-s + 1.70·14-s + 0.261·15-s − 0.947·16-s + 0.149·17-s − 0.642·18-s + 0.147·19-s + 0.0222·20-s − 1.01·21-s + 0.845·22-s − 0.639·23-s + 0.597·24-s + 0.200·25-s − 0.126·26-s + 0.968·27-s − 0.0870·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.170462843\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.170462843\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 - 1.37T + 2T^{2} \) |
| 3 | \( 1 + 1.01T + 3T^{2} \) |
| 7 | \( 1 - 4.61T + 7T^{2} \) |
| 11 | \( 1 - 2.87T + 11T^{2} \) |
| 13 | \( 1 + 0.468T + 13T^{2} \) |
| 17 | \( 1 - 0.617T + 17T^{2} \) |
| 19 | \( 1 - 0.641T + 19T^{2} \) |
| 23 | \( 1 + 3.06T + 23T^{2} \) |
| 29 | \( 1 - 6.80T + 29T^{2} \) |
| 31 | \( 1 - 2.71T + 31T^{2} \) |
| 37 | \( 1 + 3.86T + 37T^{2} \) |
| 41 | \( 1 - 1.47T + 41T^{2} \) |
| 43 | \( 1 - 0.907T + 43T^{2} \) |
| 47 | \( 1 - 5.21T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 - 3.90T + 59T^{2} \) |
| 61 | \( 1 + 8.00T + 61T^{2} \) |
| 67 | \( 1 - 5.52T + 67T^{2} \) |
| 71 | \( 1 - 1.79T + 71T^{2} \) |
| 73 | \( 1 - 15.9T + 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 + 4.36T + 83T^{2} \) |
| 89 | \( 1 - 4.45T + 89T^{2} \) |
| 97 | \( 1 - 18.1T + 97T^{2} \) |
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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
−8.817622412213467974716271494966, −8.499109655491197130042387820452, −7.56998903656393427352726639029, −6.51429283765289469641311121478, −5.73036166022269516056517798943, −4.99647118971049297458518160620, −4.46242363031410076561722858394, −3.62536157458430418093817092317, −2.39277806642274525123613372081, −0.905006276731322693150628796612,
0.905006276731322693150628796612, 2.39277806642274525123613372081, 3.62536157458430418093817092317, 4.46242363031410076561722858394, 4.99647118971049297458518160620, 5.73036166022269516056517798943, 6.51429283765289469641311121478, 7.56998903656393427352726639029, 8.499109655491197130042387820452, 8.817622412213467974716271494966
