| L(s) = 1 | + 2.94·3-s − 3.26·5-s + 4.50·7-s + 5.68·9-s − 11-s + 5.39·13-s − 9.62·15-s + 3.19·17-s + 19-s + 13.2·21-s − 1.01·23-s + 5.68·25-s + 7.90·27-s + 4.22·29-s − 3.59·31-s − 2.94·33-s − 14.7·35-s − 9.26·37-s + 15.8·39-s − 9.49·41-s + 9.64·43-s − 18.5·45-s − 10.3·47-s + 13.2·49-s + 9.41·51-s − 4.95·53-s + 3.26·55-s + ⋯ |
| L(s) = 1 | + 1.70·3-s − 1.46·5-s + 1.70·7-s + 1.89·9-s − 0.301·11-s + 1.49·13-s − 2.48·15-s + 0.775·17-s + 0.229·19-s + 2.89·21-s − 0.212·23-s + 1.13·25-s + 1.52·27-s + 0.783·29-s − 0.644·31-s − 0.512·33-s − 2.48·35-s − 1.52·37-s + 2.54·39-s − 1.48·41-s + 1.47·43-s − 2.76·45-s − 1.50·47-s + 1.89·49-s + 1.31·51-s − 0.681·53-s + 0.440·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.658639282\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.658639282\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 2.94T + 3T^{2} \) |
| 5 | \( 1 + 3.26T + 5T^{2} \) |
| 7 | \( 1 - 4.50T + 7T^{2} \) |
| 13 | \( 1 - 5.39T + 13T^{2} \) |
| 17 | \( 1 - 3.19T + 17T^{2} \) |
| 23 | \( 1 + 1.01T + 23T^{2} \) |
| 29 | \( 1 - 4.22T + 29T^{2} \) |
| 31 | \( 1 + 3.59T + 31T^{2} \) |
| 37 | \( 1 + 9.26T + 37T^{2} \) |
| 41 | \( 1 + 9.49T + 41T^{2} \) |
| 43 | \( 1 - 9.64T + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 + 4.95T + 53T^{2} \) |
| 59 | \( 1 + 2.15T + 59T^{2} \) |
| 61 | \( 1 - 6.38T + 61T^{2} \) |
| 67 | \( 1 - 5.37T + 67T^{2} \) |
| 71 | \( 1 - 6.48T + 71T^{2} \) |
| 73 | \( 1 - 14.4T + 73T^{2} \) |
| 79 | \( 1 + 16.5T + 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 - 7.92T + 89T^{2} \) |
| 97 | \( 1 + 2.72T + 97T^{2} \) |
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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
−8.505522555511222218503528699142, −7.922065218452907172290224140940, −7.68664377830675471093340161684, −6.74955865707943008736172506244, −5.30320596199948629123427254113, −4.52533193933686915062743443921, −3.66552387152147528298656392047, −3.35055381531440227400464055257, −2.04215450532832161273446232016, −1.17985019410955671794328576845,
1.17985019410955671794328576845, 2.04215450532832161273446232016, 3.35055381531440227400464055257, 3.66552387152147528298656392047, 4.52533193933686915062743443921, 5.30320596199948629123427254113, 6.74955865707943008736172506244, 7.68664377830675471093340161684, 7.922065218452907172290224140940, 8.505522555511222218503528699142
