| L(s) = 1 | − 2.20·2-s − 3·3-s − 3.13·4-s + 6.61·6-s + 7·7-s + 24.5·8-s + 9·9-s + 56.2·11-s + 9.39·12-s + 38.9·13-s − 15.4·14-s − 29.1·16-s − 119.·17-s − 19.8·18-s − 13.0·19-s − 21·21-s − 124.·22-s − 130.·23-s − 73.6·24-s − 85.8·26-s − 27·27-s − 21.9·28-s + 77.9·29-s + 61.0·31-s − 132.·32-s − 168.·33-s + 263.·34-s + ⋯ |
| L(s) = 1 | − 0.780·2-s − 0.577·3-s − 0.391·4-s + 0.450·6-s + 0.377·7-s + 1.08·8-s + 0.333·9-s + 1.54·11-s + 0.225·12-s + 0.829·13-s − 0.294·14-s − 0.455·16-s − 1.70·17-s − 0.260·18-s − 0.157·19-s − 0.218·21-s − 1.20·22-s − 1.18·23-s − 0.626·24-s − 0.647·26-s − 0.192·27-s − 0.147·28-s + 0.499·29-s + 0.353·31-s − 0.730·32-s − 0.889·33-s + 1.32·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9489113871\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9489113871\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 2 | \( 1 + 2.20T + 8T^{2} \) |
| 11 | \( 1 - 56.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 38.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 119.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 13.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 130.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 77.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 61.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 167.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 436.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 393.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 365.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 282.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 414.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 563.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 395.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 103.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 128.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 641.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 512.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.22e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 186.T + 9.12e5T^{2} \) |
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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
−10.44879076147753916713308395871, −9.412637082732924794228313163830, −8.815393012403905461297786742010, −7.973013478736682875688377135814, −6.77757053987049329868599465794, −6.03504191458922841385725447513, −4.56207152204041803999835313809, −4.00670514305580666937731514627, −1.84286148409124367116133162101, −0.72886461652620949289329986726,
0.72886461652620949289329986726, 1.84286148409124367116133162101, 4.00670514305580666937731514627, 4.56207152204041803999835313809, 6.03504191458922841385725447513, 6.77757053987049329868599465794, 7.973013478736682875688377135814, 8.815393012403905461297786742010, 9.412637082732924794228313163830, 10.44879076147753916713308395871
