| L(s) = 1 | + 2-s − 3-s + 4-s − 0.268·5-s − 6-s + 1.12·7-s + 8-s + 9-s − 0.268·10-s + 11-s − 12-s + 4.00·13-s + 1.12·14-s + 0.268·15-s + 16-s + 0.313·17-s + 18-s + 6.39·19-s − 0.268·20-s − 1.12·21-s + 22-s + 4.37·23-s − 24-s − 4.92·25-s + 4.00·26-s − 27-s + 1.12·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.120·5-s − 0.408·6-s + 0.423·7-s + 0.353·8-s + 0.333·9-s − 0.0849·10-s + 0.301·11-s − 0.288·12-s + 1.11·13-s + 0.299·14-s + 0.0693·15-s + 0.250·16-s + 0.0761·17-s + 0.235·18-s + 1.46·19-s − 0.0600·20-s − 0.244·21-s + 0.213·22-s + 0.912·23-s − 0.204·24-s − 0.985·25-s + 0.785·26-s − 0.192·27-s + 0.211·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.957823285\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.957823285\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| good | 5 | \( 1 + 0.268T + 5T^{2} \) |
| 7 | \( 1 - 1.12T + 7T^{2} \) |
| 13 | \( 1 - 4.00T + 13T^{2} \) |
| 17 | \( 1 - 0.313T + 17T^{2} \) |
| 19 | \( 1 - 6.39T + 19T^{2} \) |
| 23 | \( 1 - 4.37T + 23T^{2} \) |
| 29 | \( 1 - 2.43T + 29T^{2} \) |
| 31 | \( 1 + 2.73T + 31T^{2} \) |
| 37 | \( 1 + 7.78T + 37T^{2} \) |
| 41 | \( 1 + 6.69T + 41T^{2} \) |
| 43 | \( 1 - 4.12T + 43T^{2} \) |
| 47 | \( 1 + 1.90T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 + 7.81T + 59T^{2} \) |
| 67 | \( 1 + 7.96T + 67T^{2} \) |
| 71 | \( 1 - 14.5T + 71T^{2} \) |
| 73 | \( 1 - 15.4T + 73T^{2} \) |
| 79 | \( 1 + 3.56T + 79T^{2} \) |
| 83 | \( 1 - 15.0T + 83T^{2} \) |
| 89 | \( 1 - 13.9T + 89T^{2} \) |
| 97 | \( 1 + 6.04T + 97T^{2} \) |
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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
−8.319731370294930116525048060223, −7.54685955608561577059487098729, −6.84531637306583310124062637371, −6.13232536254204164896860193338, −5.35079246822432678489995774049, −4.86805007066755470122665107195, −3.79002237117560914939173803414, −3.28999834391476005025175943919, −1.89759716170739343492373464596, −0.975816149364099101108131808012,
0.975816149364099101108131808012, 1.89759716170739343492373464596, 3.28999834391476005025175943919, 3.79002237117560914939173803414, 4.86805007066755470122665107195, 5.35079246822432678489995774049, 6.13232536254204164896860193338, 6.84531637306583310124062637371, 7.54685955608561577059487098729, 8.319731370294930116525048060223
