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 | \( 1 + 147.T + 9.40e3T^{2} \) |
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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
−13.62874437820227864478017853605, −12.21606903046156223819738001859, −10.82297637738347456407043949204, −10.05650416436588479443667124823, −8.818619601129643290319262408583, −7.69774462672846434441974684146, −6.91679002278921590197432124933, −5.41438502334537938249381724004, −4.59550937691774711533506257286, −1.99755064699342348727652420379,
1.16910553672279480192056533942, 2.68908430737767456604877369374, 4.77169294591676266083162836541, 5.33468593298165074391683364621, 7.73732915790387707471290443383, 8.518153467704726592213929375455, 9.500419456951288968648937582619, 10.80817434832164461041429531734, 11.30108994339983195953342925105, 12.55208255955021097435234882244