| L(s) = 1 | − 3-s + 1.41·5-s + 9-s + 4·11-s − 1.41·13-s − 1.41·15-s + 4·19-s + 5.65·23-s − 2.99·25-s − 27-s − 7.07·29-s + 5.65·31-s − 4·33-s + 4.24·37-s + 1.41·39-s + 12·43-s + 1.41·45-s − 11.3·47-s − 7·49-s + 1.41·53-s + 5.65·55-s − 4·57-s − 4·59-s + 12.7·61-s − 2.00·65-s + 4·67-s − 5.65·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.632·5-s + 0.333·9-s + 1.20·11-s − 0.392·13-s − 0.365·15-s + 0.917·19-s + 1.17·23-s − 0.599·25-s − 0.192·27-s − 1.31·29-s + 1.01·31-s − 0.696·33-s + 0.697·37-s + 0.226·39-s + 1.82·43-s + 0.210·45-s − 1.65·47-s − 49-s + 0.194·53-s + 0.762·55-s − 0.529·57-s − 0.520·59-s + 1.62·61-s − 0.248·65-s + 0.488·67-s − 0.681·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.897430975\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.897430975\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| good | 5 | \( 1 - 1.41T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 1.41T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 5.65T + 23T^{2} \) |
| 29 | \( 1 + 7.07T + 29T^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 - 4.24T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 12T + 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 - 1.41T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 5.65T + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 + 16.9T + 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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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
−8.912340809376685920565184639859, −7.82892778472090565732483082008, −7.09552498117630058751411770571, −6.37650661371691200799841998257, −5.71738308704119989578794978738, −4.95100287001791700218942316770, −4.08751192369906135316933416034, −3.08930629386296579147451445934, −1.89230835385193397484900497424, −0.905767971146153151001863423843,
0.905767971146153151001863423843, 1.89230835385193397484900497424, 3.08930629386296579147451445934, 4.08751192369906135316933416034, 4.95100287001791700218942316770, 5.71738308704119989578794978738, 6.37650661371691200799841998257, 7.09552498117630058751411770571, 7.82892778472090565732483082008, 8.912340809376685920565184639859
