L(s) = 1 | + (0.866 − 0.5i)3-s + (−0.5 − 0.866i)4-s + (−0.866 + 0.5i)7-s + (0.499 − 0.866i)9-s + (−0.866 − 0.499i)12-s + (−0.866 − 1.5i)13-s + (−0.499 + 0.866i)16-s + (−0.499 + 0.866i)21-s − 0.999i·27-s + (0.866 + 0.499i)28-s − 1.73i·31-s − 0.999·36-s − i·37-s + (−1.5 − 0.866i)39-s − 1.73·43-s + ⋯ |
L(s) = 1 | + (0.866 − 0.5i)3-s + (−0.5 − 0.866i)4-s + (−0.866 + 0.5i)7-s + (0.499 − 0.866i)9-s + (−0.866 − 0.499i)12-s + (−0.866 − 1.5i)13-s + (−0.499 + 0.866i)16-s + (−0.499 + 0.866i)21-s − 0.999i·27-s + (0.866 + 0.499i)28-s − 1.73i·31-s − 0.999·36-s − i·37-s + (−1.5 − 0.866i)39-s − 1.73·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.769 + 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.769 + 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9970403614\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9970403614\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 5 | \( 1 \) |
| 37 | \( 1 + iT \) |
good | 2 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + 1.73iT - T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + 1.73T + T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 - iT - T^{2} \) |
| 79 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 - 1.73T + T^{2} \) |
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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
−8.774185336426344890259972030387, −7.995890017133682802889822601042, −7.28965200022887898801645671932, −6.30658264706871980479715539672, −5.75477604392080847559428312059, −4.85326081106356257485626276436, −3.75135460286382860063789212531, −2.87266190732957214268904877643, −2.02348480493647955834351647462, −0.54878027015358715741982972381,
1.92718824442053268999928333034, 3.08924180433997133229528868416, 3.55876433443509702611320613231, 4.52311464367830147949812882282, 4.96105797523618330106837590952, 6.62166500434896194511028762739, 7.05928996303693308676517763204, 7.87921110668924798582396507562, 8.674311308451422482992388388874, 9.204701853059893237358321185755