| L(s) = 1 | + 2-s + 3-s + 4-s − 1.65·5-s + 6-s − 1.05·7-s + 8-s + 9-s − 1.65·10-s − 2.77·11-s + 12-s + 5.36·13-s − 1.05·14-s − 1.65·15-s + 16-s + 5.11·17-s + 18-s − 0.230·19-s − 1.65·20-s − 1.05·21-s − 2.77·22-s + 23-s + 24-s − 2.24·25-s + 5.36·26-s + 27-s − 1.05·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.741·5-s + 0.408·6-s − 0.399·7-s + 0.353·8-s + 0.333·9-s − 0.524·10-s − 0.836·11-s + 0.288·12-s + 1.48·13-s − 0.282·14-s − 0.428·15-s + 0.250·16-s + 1.24·17-s + 0.235·18-s − 0.0528·19-s − 0.370·20-s − 0.230·21-s − 0.591·22-s + 0.208·23-s + 0.204·24-s − 0.449·25-s + 1.05·26-s + 0.192·27-s − 0.199·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.332686705\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.332686705\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 + 1.65T + 5T^{2} \) |
| 7 | \( 1 + 1.05T + 7T^{2} \) |
| 11 | \( 1 + 2.77T + 11T^{2} \) |
| 13 | \( 1 - 5.36T + 13T^{2} \) |
| 17 | \( 1 - 5.11T + 17T^{2} \) |
| 19 | \( 1 + 0.230T + 19T^{2} \) |
| 31 | \( 1 - 3.97T + 31T^{2} \) |
| 37 | \( 1 + 4.94T + 37T^{2} \) |
| 41 | \( 1 - 5.80T + 41T^{2} \) |
| 43 | \( 1 - 2.66T + 43T^{2} \) |
| 47 | \( 1 - 4.87T + 47T^{2} \) |
| 53 | \( 1 - 5.26T + 53T^{2} \) |
| 59 | \( 1 - 6.68T + 59T^{2} \) |
| 61 | \( 1 - 1.45T + 61T^{2} \) |
| 67 | \( 1 + 7.07T + 67T^{2} \) |
| 71 | \( 1 - 3.51T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 + 2.10T + 79T^{2} \) |
| 83 | \( 1 + 1.00T + 83T^{2} \) |
| 89 | \( 1 - 11.2T + 89T^{2} \) |
| 97 | \( 1 - 0.785T + 97T^{2} \) |
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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
−8.233050067656247049022802401632, −7.77588549798837505758050066715, −7.04870718222913705343404270088, −6.11847650518150485308171142123, −5.50250020839375996009761349117, −4.50934684592448873533060077993, −3.63254390445407893727521759189, −3.30046838257469907063367438776, −2.24591889743358404158730500538, −0.938313745523045900913261984243,
0.938313745523045900913261984243, 2.24591889743358404158730500538, 3.30046838257469907063367438776, 3.63254390445407893727521759189, 4.50934684592448873533060077993, 5.50250020839375996009761349117, 6.11847650518150485308171142123, 7.04870718222913705343404270088, 7.77588549798837505758050066715, 8.233050067656247049022802401632
