| L(s) = 1 | − 3-s + 2.70·5-s − 4.70·7-s + 9-s + 4.70·11-s + 6·13-s − 2.70·15-s − 2.70·17-s + 19-s + 4.70·21-s + 4·23-s + 2.29·25-s − 27-s + 2·29-s + 9.40·31-s − 4.70·33-s − 12.7·35-s − 3.40·37-s − 6·39-s − 3.40·41-s + 10.1·43-s + 2.70·45-s − 0.701·47-s + 15.1·49-s + 2.70·51-s − 6·53-s + 12.7·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.20·5-s − 1.77·7-s + 0.333·9-s + 1.41·11-s + 1.66·13-s − 0.697·15-s − 0.655·17-s + 0.229·19-s + 1.02·21-s + 0.834·23-s + 0.459·25-s − 0.192·27-s + 0.371·29-s + 1.68·31-s − 0.818·33-s − 2.14·35-s − 0.559·37-s − 0.960·39-s − 0.531·41-s + 1.54·43-s + 0.402·45-s − 0.102·47-s + 2.15·49-s + 0.378·51-s − 0.824·53-s + 1.71·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.356146331\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.356146331\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 2.70T + 5T^{2} \) |
| 7 | \( 1 + 4.70T + 7T^{2} \) |
| 11 | \( 1 - 4.70T + 11T^{2} \) |
| 13 | \( 1 - 6T + 13T^{2} \) |
| 17 | \( 1 + 2.70T + 17T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 9.40T + 31T^{2} \) |
| 37 | \( 1 + 3.40T + 37T^{2} \) |
| 41 | \( 1 + 3.40T + 41T^{2} \) |
| 43 | \( 1 - 10.1T + 43T^{2} \) |
| 47 | \( 1 + 0.701T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 1.29T + 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 6.70T + 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 + 12.8T + 89T^{2} \) |
| 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
−10.96769913715293099475192066968, −10.09176710965333885268304108732, −9.362769571212070527950979918148, −8.757313813718085837399401249487, −6.81206405484763842093853441835, −6.36366132779300823458241410118, −5.79862135858770537834463642990, −4.18380000884211207195105248698, −3.02584527309603030292322932266, −1.24563706582482003593720389702,
1.24563706582482003593720389702, 3.02584527309603030292322932266, 4.18380000884211207195105248698, 5.79862135858770537834463642990, 6.36366132779300823458241410118, 6.81206405484763842093853441835, 8.757313813718085837399401249487, 9.362769571212070527950979918148, 10.09176710965333885268304108732, 10.96769913715293099475192066968
