L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.866 + 0.5i)3-s + (−0.499 + 0.866i)4-s + (−0.866 − 0.499i)6-s + (−2.56 + 4.44i)7-s − 0.999·8-s + (0.499 − 0.866i)9-s + (−4.83 + 2.78i)11-s − 0.999i·12-s + (1.67 + 3.19i)13-s − 5.13·14-s + (−0.5 − 0.866i)16-s + (1.11 + 0.641i)17-s + 0.999·18-s + (−5.75 − 3.32i)19-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.499 + 0.288i)3-s + (−0.249 + 0.433i)4-s + (−0.353 − 0.204i)6-s + (−0.969 + 1.67i)7-s − 0.353·8-s + (0.166 − 0.288i)9-s + (−1.45 + 0.841i)11-s − 0.288i·12-s + (0.465 + 0.884i)13-s − 1.37·14-s + (−0.125 − 0.216i)16-s + (0.269 + 0.155i)17-s + 0.235·18-s + (−1.32 − 0.762i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00792 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00792 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4589067135\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4589067135\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + (0.866 - 0.5i)T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (-1.67 - 3.19i)T \) |
good | 7 | \( 1 + (2.56 - 4.44i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (4.83 - 2.78i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-1.11 - 0.641i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (5.75 + 3.32i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.94 + 3.43i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.75 - 3.03i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 6.30iT - 31T^{2} \) |
| 37 | \( 1 + (4.19 + 7.25i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.11 + 0.641i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.77 - 2.17i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 5.02T + 47T^{2} \) |
| 53 | \( 1 - 6.30iT - 53T^{2} \) |
| 59 | \( 1 + (1.31 + 0.759i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.19 + 8.99i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.43 + 2.48i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-9.44 - 5.45i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 4.94T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 + 17.3T + 83T^{2} \) |
| 89 | \( 1 + (5.57 - 3.22i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (3.58 - 6.20i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.557453843779359615144973440428, −8.902694114632810558851108446231, −8.404894095327981359386903660793, −7.03519537292456738983534326116, −6.62651890400923602011366255025, −5.68970814476815810435656935481, −5.14593932461456808675895363361, −4.32176889949948621426564636776, −3.02022963718719308505100456462, −2.27068196708121805012969268817,
0.17846796126112086218603635251, 1.05793775283766833374277192874, 2.71728602621848592657818296353, 3.50504448781936422557518841721, 4.32782796015551220053614579343, 5.43939397919703652792983741271, 6.05448868862248270428318572309, 6.96520829221149579421199959482, 7.76791253920293614598884349857, 8.491948869995501501705695494653