L(s) = 1 | − 2-s − 2.19·3-s + 4-s + 0.174·5-s + 2.19·6-s + 3.49·7-s − 8-s + 1.82·9-s − 0.174·10-s − 0.405·11-s − 2.19·12-s + 4.26·13-s − 3.49·14-s − 0.383·15-s + 16-s − 2.55·17-s − 1.82·18-s + 4.99·19-s + 0.174·20-s − 7.67·21-s + 0.405·22-s − 23-s + 2.19·24-s − 4.96·25-s − 4.26·26-s + 2.57·27-s + 3.49·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.26·3-s + 0.5·4-s + 0.0781·5-s + 0.896·6-s + 1.31·7-s − 0.353·8-s + 0.608·9-s − 0.0552·10-s − 0.122·11-s − 0.634·12-s + 1.18·13-s − 0.933·14-s − 0.0991·15-s + 0.250·16-s − 0.618·17-s − 0.430·18-s + 1.14·19-s + 0.0390·20-s − 1.67·21-s + 0.0864·22-s − 0.208·23-s + 0.448·24-s − 0.993·25-s − 0.836·26-s + 0.496·27-s + 0.659·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.131748960\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.131748960\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 - T \) |
good | 3 | \( 1 + 2.19T + 3T^{2} \) |
| 5 | \( 1 - 0.174T + 5T^{2} \) |
| 7 | \( 1 - 3.49T + 7T^{2} \) |
| 11 | \( 1 + 0.405T + 11T^{2} \) |
| 13 | \( 1 - 4.26T + 13T^{2} \) |
| 17 | \( 1 + 2.55T + 17T^{2} \) |
| 19 | \( 1 - 4.99T + 19T^{2} \) |
| 29 | \( 1 - 4.57T + 29T^{2} \) |
| 31 | \( 1 - 1.01T + 31T^{2} \) |
| 37 | \( 1 - 2.11T + 37T^{2} \) |
| 41 | \( 1 - 6.75T + 41T^{2} \) |
| 43 | \( 1 + 3.39T + 43T^{2} \) |
| 47 | \( 1 - 9.79T + 47T^{2} \) |
| 53 | \( 1 + 4.54T + 53T^{2} \) |
| 59 | \( 1 + 7.05T + 59T^{2} \) |
| 61 | \( 1 - 7.13T + 61T^{2} \) |
| 67 | \( 1 + 1.88T + 67T^{2} \) |
| 71 | \( 1 - 6.82T + 71T^{2} \) |
| 73 | \( 1 - 16.3T + 73T^{2} \) |
| 79 | \( 1 + 9.05T + 79T^{2} \) |
| 83 | \( 1 - 3.41T + 83T^{2} \) |
| 89 | \( 1 - 8.83T + 89T^{2} \) |
| 97 | \( 1 - 13.4T + 97T^{2} \) |
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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
−8.043952047791780479786708328919, −7.49332008522277023713341740958, −6.54150171432426303325984785728, −6.00938548646195383364897355726, −5.33411734085911217699814543027, −4.67471111404436985377733384428, −3.76172218452729682708586809234, −2.48821113455409957467344741407, −1.45289373943785358234996989289, −0.73431558452057058828141016693,
0.73431558452057058828141016693, 1.45289373943785358234996989289, 2.48821113455409957467344741407, 3.76172218452729682708586809234, 4.67471111404436985377733384428, 5.33411734085911217699814543027, 6.00938548646195383364897355726, 6.54150171432426303325984785728, 7.49332008522277023713341740958, 8.043952047791780479786708328919