L(s) = 1 | + (−0.866 + 1.5i)3-s + (−1.73 + i)4-s + (−1.13 + 4.23i)7-s + (−1.5 − 2.59i)9-s − 3.46i·12-s + (1.99 − 3.46i)16-s + (−3.09 − 0.830i)19-s + (−5.36 − 5.36i)21-s − 5i·25-s + 5.19·27-s + (−2.26 − 8.46i)28-s + (−0.830 + 0.830i)31-s + (5.19 + 3i)36-s + (−11.5 + 3.09i)37-s + (1.5 − 0.866i)43-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.866i)3-s + (−0.866 + 0.5i)4-s + (−0.428 + 1.59i)7-s + (−0.5 − 0.866i)9-s − 0.999i·12-s + (0.499 − 0.866i)16-s + (−0.710 − 0.190i)19-s + (−1.17 − 1.17i)21-s − i·25-s + 1.00·27-s + (−0.428 − 1.59i)28-s + (−0.149 + 0.149i)31-s + (0.866 + 0.5i)36-s + (−1.90 + 0.509i)37-s + (0.228 − 0.132i)43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.720 + 0.693i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.720 + 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.113834 - 0.282532i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.113834 - 0.282532i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.866 - 1.5i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (1.73 - i)T^{2} \) |
| 5 | \( 1 + 5iT^{2} \) |
| 7 | \( 1 + (1.13 - 4.23i)T + (-6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (-9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.09 + 0.830i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (0.830 - 0.830i)T - 31iT^{2} \) |
| 37 | \( 1 + (11.5 - 3.09i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (-1.5 + 0.866i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 47iT^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + (51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (4.33 + 7.5i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.23 - 15.7i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (-61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (-7.63 - 7.63i)T + 73iT^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 + 83iT^{2} \) |
| 89 | \( 1 + (-77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (9.59 + 2.57i)T + (84.0 + 48.5i)T^{2} \) |
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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
−11.60563517437656023725314236624, −10.41476516339243093075877336786, −9.602784963727829802565887663610, −8.827083227970116028318788330201, −8.361843631936297513570883215704, −6.67278585731478039800933820982, −5.65585671774579207312841732209, −4.91809626670092638930892445213, −3.82135952072599015126685887052, −2.70460698038876625813352243335,
0.20211704338225197482190287919, 1.52052355798266844546797486705, 3.59663957312564844909724838840, 4.66501686846975105320051501208, 5.72362514744095755516551829282, 6.73367803584789186798194326797, 7.47996980058619059698468913206, 8.490014850921239061565827719658, 9.581915500000721342040403284060, 10.55323092677649736096849064675