L(s) = 1 | + 2-s − 2.23·3-s + 4-s + 1.35·5-s − 2.23·6-s + 1.10·7-s + 8-s + 1.99·9-s + 1.35·10-s − 4.91·11-s − 2.23·12-s − 6.70·13-s + 1.10·14-s − 3.01·15-s + 16-s + 2.39·17-s + 1.99·18-s − 2.31·19-s + 1.35·20-s − 2.47·21-s − 4.91·22-s + 1.62·23-s − 2.23·24-s − 3.17·25-s − 6.70·26-s + 2.25·27-s + 1.10·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.28·3-s + 0.5·4-s + 0.604·5-s − 0.912·6-s + 0.419·7-s + 0.353·8-s + 0.663·9-s + 0.427·10-s − 1.48·11-s − 0.644·12-s − 1.85·13-s + 0.296·14-s − 0.779·15-s + 0.250·16-s + 0.581·17-s + 0.469·18-s − 0.531·19-s + 0.302·20-s − 0.541·21-s − 1.04·22-s + 0.338·23-s − 0.456·24-s − 0.635·25-s − 1.31·26-s + 0.433·27-s + 0.209·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.624180053\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.624180053\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 2003 | \( 1 + T \) |
good | 3 | \( 1 + 2.23T + 3T^{2} \) |
| 5 | \( 1 - 1.35T + 5T^{2} \) |
| 7 | \( 1 - 1.10T + 7T^{2} \) |
| 11 | \( 1 + 4.91T + 11T^{2} \) |
| 13 | \( 1 + 6.70T + 13T^{2} \) |
| 17 | \( 1 - 2.39T + 17T^{2} \) |
| 19 | \( 1 + 2.31T + 19T^{2} \) |
| 23 | \( 1 - 1.62T + 23T^{2} \) |
| 29 | \( 1 + 1.50T + 29T^{2} \) |
| 31 | \( 1 - 8.01T + 31T^{2} \) |
| 37 | \( 1 + 0.245T + 37T^{2} \) |
| 41 | \( 1 - 9.41T + 41T^{2} \) |
| 43 | \( 1 - 8.36T + 43T^{2} \) |
| 47 | \( 1 - 4.54T + 47T^{2} \) |
| 53 | \( 1 - 13.6T + 53T^{2} \) |
| 59 | \( 1 + 5.93T + 59T^{2} \) |
| 61 | \( 1 + 0.490T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 - 1.78T + 71T^{2} \) |
| 73 | \( 1 - 15.0T + 73T^{2} \) |
| 79 | \( 1 - 11.7T + 79T^{2} \) |
| 83 | \( 1 + 3.02T + 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 + 5.97T + 97T^{2} \) |
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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
−8.120253223471464529498628064134, −7.56707992603483783510498345612, −6.80056165544362250449103079928, −5.92907714771868409608853325943, −5.40069807654659150629422433644, −4.96581035961868890582878144570, −4.24347274970708576468747596159, −2.71816197170543284035338073590, −2.23882985039917251960788975853, −0.67213653215231851047333980969,
0.67213653215231851047333980969, 2.23882985039917251960788975853, 2.71816197170543284035338073590, 4.24347274970708576468747596159, 4.96581035961868890582878144570, 5.40069807654659150629422433644, 5.92907714771868409608853325943, 6.80056165544362250449103079928, 7.56707992603483783510498345612, 8.120253223471464529498628064134