L(s) = 1 | + (−0.455 − 1.94i)2-s + (−3.58 + 1.77i)4-s + (3.40 − 3.40i)5-s + 12.1·7-s + (5.08 + 6.17i)8-s + (−8.18 − 5.08i)10-s + (−9.81 − 9.81i)11-s + (−7.76 − 7.76i)13-s + (−5.51 − 23.6i)14-s + (9.71 − 12.7i)16-s − 9.73·17-s + (11.2 − 11.2i)19-s + (−6.17 + 18.2i)20-s + (−14.6 + 23.5i)22-s + 20.2·23-s + ⋯ |
L(s) = 1 | + (−0.227 − 0.973i)2-s + (−0.896 + 0.443i)4-s + (0.681 − 0.681i)5-s + 1.73·7-s + (0.635 + 0.772i)8-s + (−0.818 − 0.508i)10-s + (−0.891 − 0.891i)11-s + (−0.597 − 0.597i)13-s + (−0.394 − 1.68i)14-s + (0.607 − 0.794i)16-s − 0.572·17-s + (0.593 − 0.593i)19-s + (−0.308 + 0.912i)20-s + (−0.665 + 1.07i)22-s + 0.881·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.257 + 0.966i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.257 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.864805 - 1.12495i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.864805 - 1.12495i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.455 + 1.94i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-3.40 + 3.40i)T - 25iT^{2} \) |
| 7 | \( 1 - 12.1T + 49T^{2} \) |
| 11 | \( 1 + (9.81 + 9.81i)T + 121iT^{2} \) |
| 13 | \( 1 + (7.76 + 7.76i)T + 169iT^{2} \) |
| 17 | \( 1 + 9.73T + 289T^{2} \) |
| 19 | \( 1 + (-11.2 + 11.2i)T - 361iT^{2} \) |
| 23 | \( 1 - 20.2T + 529T^{2} \) |
| 29 | \( 1 + (-16.4 - 16.4i)T + 841iT^{2} \) |
| 31 | \( 1 + 26.3iT - 961T^{2} \) |
| 37 | \( 1 + (23.7 - 23.7i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 - 24.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (-29.8 - 29.8i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 - 31.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (36.8 - 36.8i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + (-14.1 - 14.1i)T + 3.48e3iT^{2} \) |
| 61 | \( 1 + (42.5 + 42.5i)T + 3.72e3iT^{2} \) |
| 67 | \( 1 + (-48.7 + 48.7i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 + 7.73T + 5.04e3T^{2} \) |
| 73 | \( 1 - 85.4iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 105. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-62.1 + 62.1i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 - 127. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 147.T + 9.40e3T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.55208255955021097435234882244, −11.30108994339983195953342925105, −10.80817434832164461041429531734, −9.500419456951288968648937582619, −8.518153467704726592213929375455, −7.73732915790387707471290443383, −5.33468593298165074391683364621, −4.77169294591676266083162836541, −2.68908430737767456604877369374, −1.16910553672279480192056533942,
1.99755064699342348727652420379, 4.59550937691774711533506257286, 5.41438502334537938249381724004, 6.91679002278921590197432124933, 7.69774462672846434441974684146, 8.818619601129643290319262408583, 10.05650416436588479443667124823, 10.82297637738347456407043949204, 12.21606903046156223819738001859, 13.62874437820227864478017853605