| L(s) = 1 | − 2-s − 3-s + 4-s − 1.73·5-s + 6-s + 4.73·7-s − 8-s + 9-s + 1.73·10-s + 4.73·11-s − 12-s − 4.73·14-s + 1.73·15-s + 16-s − 5.19·17-s − 18-s + 1.26·19-s − 1.73·20-s − 4.73·21-s − 4.73·22-s + 2.19·23-s + 24-s − 2.00·25-s − 27-s + 4.73·28-s − 3·29-s − 1.73·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.774·5-s + 0.408·6-s + 1.78·7-s − 0.353·8-s + 0.333·9-s + 0.547·10-s + 1.42·11-s − 0.288·12-s − 1.26·14-s + 0.447·15-s + 0.250·16-s − 1.26·17-s − 0.235·18-s + 0.290·19-s − 0.387·20-s − 1.03·21-s − 1.00·22-s + 0.457·23-s + 0.204·24-s − 0.400·25-s − 0.192·27-s + 0.894·28-s − 0.557·29-s − 0.316·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.065197889\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.065197889\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 1.73T + 5T^{2} \) |
| 7 | \( 1 - 4.73T + 7T^{2} \) |
| 11 | \( 1 - 4.73T + 11T^{2} \) |
| 17 | \( 1 + 5.19T + 17T^{2} \) |
| 19 | \( 1 - 1.26T + 19T^{2} \) |
| 23 | \( 1 - 2.19T + 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 - 2.53T + 31T^{2} \) |
| 37 | \( 1 - 3T + 37T^{2} \) |
| 41 | \( 1 + 0.464T + 41T^{2} \) |
| 43 | \( 1 - 6.19T + 43T^{2} \) |
| 47 | \( 1 + 1.26T + 47T^{2} \) |
| 53 | \( 1 - 3T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 - 4.80T + 61T^{2} \) |
| 67 | \( 1 - 10.7T + 67T^{2} \) |
| 71 | \( 1 - 8.19T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 + 11.6T + 83T^{2} \) |
| 89 | \( 1 - 2.53T + 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−9.980983945840852271312885742176, −8.929598857225764831839232099103, −8.388720418098197273317052905760, −7.47439812866991414027100381471, −6.84183962971155871168906744702, −5.71893880791084169055256283817, −4.60281403229524829320668095573, −3.93617476386783947085651831135, −2.08149792552827418837165095160, −0.974560627459520518158610227161,
0.974560627459520518158610227161, 2.08149792552827418837165095160, 3.93617476386783947085651831135, 4.60281403229524829320668095573, 5.71893880791084169055256283817, 6.84183962971155871168906744702, 7.47439812866991414027100381471, 8.388720418098197273317052905760, 8.929598857225764831839232099103, 9.980983945840852271312885742176
