L(s) = 1 | + (0.866 + 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s + (−0.499 − 0.866i)6-s + (−1.73 + i)7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (−1 + 1.73i)11-s − 0.999i·12-s + (2.59 − 2.5i)13-s − 1.99·14-s + (−0.5 + 0.866i)16-s + (4.33 − 2.5i)17-s + 0.999i·18-s + (−1 − 1.73i)19-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (−0.499 − 0.288i)3-s + (0.249 + 0.433i)4-s + (−0.204 − 0.353i)6-s + (−0.654 + 0.377i)7-s + 0.353i·8-s + (0.166 + 0.288i)9-s + (−0.301 + 0.522i)11-s − 0.288i·12-s + (0.720 − 0.693i)13-s − 0.534·14-s + (−0.125 + 0.216i)16-s + (1.05 − 0.606i)17-s + 0.235i·18-s + (−0.229 − 0.397i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0342 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0342 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.722619606\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.722619606\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (-2.59 + 2.5i)T \) |
good | 7 | \( 1 + (1.73 - i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-4.33 + 2.5i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.19 - 3i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (4.5 - 7.79i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + (-9.52 - 5.5i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2.5 - 4.33i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (8.66 - 5i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 2iT - 47T^{2} \) |
| 53 | \( 1 - iT - 53T^{2} \) |
| 59 | \( 1 + (4 + 6.92i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.5 - 9.52i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.73 + i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-7 - 12.1i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 13iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + (-1 + 1.73i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.73 - i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.428550129888476964603605920622, −8.446923157207888788898746696663, −7.57250309441479910301852639783, −6.95431935113321805623160794238, −6.12682043013550900797513725545, −5.40158849775040088339015123524, −4.80150889424386822749895787386, −3.45704097557826972763104011884, −2.82617662725035052444046728681, −1.27614535732585334893225109974,
0.58527917790090619339315446663, 1.99848931585558765634756222222, 3.42981113604179270070234333560, 3.83048036204672428938971351882, 4.89775483452651950602571463866, 5.84318407051633069221411325609, 6.29637648494992533254269217621, 7.22432761620875432977219670584, 8.214066965552417896858721520709, 9.215746470781293505810015751751