L(s) = 1 | + (−1 − 1.73i)5-s + (2 − 3.46i)11-s + (1.5 − 2.59i)13-s + (−3.5 − 6.06i)17-s + (2.5 − 4.33i)19-s + (2 + 3.46i)23-s + (0.500 − 0.866i)25-s + (−0.5 − 0.866i)29-s + 3·31-s + (−5.5 + 9.52i)37-s + (4.5 − 7.79i)41-s + (−2.5 − 4.33i)43-s + 3·47-s + (1.5 + 2.59i)53-s − 7.99·55-s + ⋯ |
L(s) = 1 | + (−0.447 − 0.774i)5-s + (0.603 − 1.04i)11-s + (0.416 − 0.720i)13-s + (−0.848 − 1.47i)17-s + (0.573 − 0.993i)19-s + (0.417 + 0.722i)23-s + (0.100 − 0.173i)25-s + (−0.0928 − 0.160i)29-s + 0.538·31-s + (−0.904 + 1.56i)37-s + (0.702 − 1.21i)41-s + (−0.381 − 0.660i)43-s + 0.437·47-s + (0.206 + 0.356i)53-s − 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.841 + 0.540i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.841 + 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.499188949\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.499188949\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.5 + 2.59i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (3.5 + 6.06i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.5 + 4.33i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 + (5.5 - 9.52i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.5 + 7.79i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.5 + 4.33i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 3T + 47T^{2} \) |
| 53 | \( 1 + (-1.5 - 2.59i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 7T + 59T^{2} \) |
| 61 | \( 1 + 3T + 61T^{2} \) |
| 67 | \( 1 - 13T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + (-3.5 - 6.06i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + 9T + 79T^{2} \) |
| 83 | \( 1 + (0.5 + 0.866i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (7.5 - 12.9i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (8.5 + 14.7i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.044008778603206265035216160699, −7.13062751110142116872727820950, −6.59450139267222933618571231482, −5.53528593985476362032981996572, −5.04978739854270182573705953432, −4.22715439865998572172339636866, −3.34188629903753066842655143238, −2.62606682154743824723359755987, −1.12742308660158117373413333974, −0.45174014862334022527628124690,
1.41890205762233065849731444528, 2.20166741297343334647315815421, 3.34759211510826943030556262876, 4.02646712820557168961653096035, 4.60093741361838950831408494071, 5.75310812878798172400534994717, 6.54803649366268306762141582378, 6.91262350179843331823121520073, 7.72968583124454813501522639041, 8.425227464756000798854648875019