L(s) = 1 | + 2-s + 4-s − 1.64·5-s + 8-s − 1.64·10-s − 1.64·11-s − 0.645·13-s + 16-s + 1.64·17-s − 2·19-s − 1.64·20-s − 1.64·22-s + 9.29·23-s − 2.29·25-s − 0.645·26-s + 7.64·29-s − 0.645·31-s + 32-s + 1.64·34-s + 3.93·37-s − 2·38-s − 1.64·40-s + 4.93·41-s + 5·43-s − 1.64·44-s + 9.29·46-s + 10.9·47-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.736·5-s + 0.353·8-s − 0.520·10-s − 0.496·11-s − 0.179·13-s + 0.250·16-s + 0.399·17-s − 0.458·19-s − 0.368·20-s − 0.350·22-s + 1.93·23-s − 0.458·25-s − 0.126·26-s + 1.41·29-s − 0.115·31-s + 0.176·32-s + 0.282·34-s + 0.647·37-s − 0.324·38-s − 0.260·40-s + 0.771·41-s + 0.762·43-s − 0.248·44-s + 1.36·46-s + 1.59·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.459333532\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.459333532\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 1.64T + 5T^{2} \) |
| 11 | \( 1 + 1.64T + 11T^{2} \) |
| 13 | \( 1 + 0.645T + 13T^{2} \) |
| 17 | \( 1 - 1.64T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 9.29T + 23T^{2} \) |
| 29 | \( 1 - 7.64T + 29T^{2} \) |
| 31 | \( 1 + 0.645T + 31T^{2} \) |
| 37 | \( 1 - 3.93T + 37T^{2} \) |
| 41 | \( 1 - 4.93T + 41T^{2} \) |
| 43 | \( 1 - 5T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 13.6T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 - 8.29T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 - 10.5T + 73T^{2} \) |
| 79 | \( 1 + 15.2T + 79T^{2} \) |
| 83 | \( 1 - 2.70T + 83T^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 - 7.58T + 97T^{2} \) |
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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
−8.783201829758983090886125222924, −7.87373762664283195166597787366, −7.38019787192156454827497270862, −6.52942671909187957809797676204, −5.66966395975908589821427917253, −4.81783855596281338002717693695, −4.17757073131082453283400669081, −3.18870127220997976273791648814, −2.44581985130163910558282353412, −0.895046676809971923298890429254,
0.895046676809971923298890429254, 2.44581985130163910558282353412, 3.18870127220997976273791648814, 4.17757073131082453283400669081, 4.81783855596281338002717693695, 5.66966395975908589821427917253, 6.52942671909187957809797676204, 7.38019787192156454827497270862, 7.87373762664283195166597787366, 8.783201829758983090886125222924