| L(s) = 1 | + 2-s + 4-s − 1.61·7-s + 8-s − 0.618·11-s + 1.85·13-s − 1.61·14-s + 16-s + 5.23·17-s − 0.854·19-s − 0.618·22-s − 1.85·23-s + 1.85·26-s − 1.61·28-s + 7.23·29-s + 6.47·31-s + 32-s + 5.23·34-s − 10.5·37-s − 0.854·38-s + 11.6·41-s + 7.70·43-s − 0.618·44-s − 1.85·46-s − 0.618·47-s − 4.38·49-s + 1.85·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.611·7-s + 0.353·8-s − 0.186·11-s + 0.514·13-s − 0.432·14-s + 0.250·16-s + 1.26·17-s − 0.195·19-s − 0.131·22-s − 0.386·23-s + 0.363·26-s − 0.305·28-s + 1.34·29-s + 1.16·31-s + 0.176·32-s + 0.897·34-s − 1.73·37-s − 0.138·38-s + 1.81·41-s + 1.17·43-s − 0.0931·44-s − 0.273·46-s − 0.0901·47-s − 0.625·49-s + 0.257·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.790412210\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.790412210\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 1.61T + 7T^{2} \) |
| 11 | \( 1 + 0.618T + 11T^{2} \) |
| 13 | \( 1 - 1.85T + 13T^{2} \) |
| 17 | \( 1 - 5.23T + 17T^{2} \) |
| 19 | \( 1 + 0.854T + 19T^{2} \) |
| 23 | \( 1 + 1.85T + 23T^{2} \) |
| 29 | \( 1 - 7.23T + 29T^{2} \) |
| 31 | \( 1 - 6.47T + 31T^{2} \) |
| 37 | \( 1 + 10.5T + 37T^{2} \) |
| 41 | \( 1 - 11.6T + 41T^{2} \) |
| 43 | \( 1 - 7.70T + 43T^{2} \) |
| 47 | \( 1 + 0.618T + 47T^{2} \) |
| 53 | \( 1 - 7.61T + 53T^{2} \) |
| 59 | \( 1 - 1.90T + 59T^{2} \) |
| 61 | \( 1 - 3.70T + 61T^{2} \) |
| 67 | \( 1 + 9.70T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 - 4.94T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 - 2.94T + 83T^{2} \) |
| 89 | \( 1 - 6.90T + 89T^{2} \) |
| 97 | \( 1 - 3.70T + 97T^{2} \) |
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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
−8.994175231820274778386625090189, −8.162723106287645149574702939021, −7.40238528914668239555837480408, −6.49025152115898531835873416681, −5.91927137329491264217049462702, −5.07362236824970282601790625966, −4.10562238265728601118870706803, −3.30556872328845877159420863743, −2.45897844476634063028885193831, −1.01751205049625935798024249635,
1.01751205049625935798024249635, 2.45897844476634063028885193831, 3.30556872328845877159420863743, 4.10562238265728601118870706803, 5.07362236824970282601790625966, 5.91927137329491264217049462702, 6.49025152115898531835873416681, 7.40238528914668239555837480408, 8.162723106287645149574702939021, 8.994175231820274778386625090189
