L(s) = 1 | + (−1.87 + 3.24i)5-s + (2.64 + 4.58i)11-s − 4.24·13-s + (−1.87 − 3.24i)17-s + (−1.41 + 2.44i)19-s + (−2.64 + 4.58i)23-s + (−4.5 − 7.79i)25-s + 5.29·29-s + (−4.24 − 7.34i)31-s + (−2 + 3.46i)37-s + 3.74·41-s + 8·43-s + (3.74 − 6.48i)47-s + (−5.29 − 9.16i)53-s − 19.7·55-s + ⋯ |
L(s) = 1 | + (−0.836 + 1.44i)5-s + (0.797 + 1.38i)11-s − 1.17·13-s + (−0.453 − 0.785i)17-s + (−0.324 + 0.561i)19-s + (−0.551 + 0.955i)23-s + (−0.900 − 1.55i)25-s + 0.982·29-s + (−0.762 − 1.31i)31-s + (−0.328 + 0.569i)37-s + 0.584·41-s + 1.21·43-s + (0.545 − 0.945i)47-s + (−0.726 − 1.25i)53-s − 2.66·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 + 0.435i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.900 + 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4370444427\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4370444427\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (1.87 - 3.24i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.64 - 4.58i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 4.24T + 13T^{2} \) |
| 17 | \( 1 + (1.87 + 3.24i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.41 - 2.44i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.64 - 4.58i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 5.29T + 29T^{2} \) |
| 31 | \( 1 + (4.24 + 7.34i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2 - 3.46i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 3.74T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + (-3.74 + 6.48i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (5.29 + 9.16i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.74 - 6.48i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4.94 - 8.57i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6 + 10.3i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 15.8T + 71T^{2} \) |
| 73 | \( 1 + (0.707 + 1.22i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 14.9T + 83T^{2} \) |
| 89 | \( 1 + (-1.87 + 3.24i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 9.89T + 97T^{2} \) |
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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
−9.866244995066786610370335283408, −9.111583969260647815952604972247, −7.81832660761708586465171601211, −7.30759444260070199141719317822, −6.86651681836844527752836040415, −5.87984296001266987165570757462, −4.56490620446982710238071242104, −3.97910243464005750895518838363, −2.87295126802880146497338559469, −2.01359059936647943737709340375,
0.16934341525545764998152237659, 1.29793382993317383663837591469, 2.82407687156100621222680710686, 4.08000232691590478345869395842, 4.51934704359954971712774087280, 5.51816222210376724283882441058, 6.39136888184674163482025564547, 7.40998116265851159516729250429, 8.266967664373551941343891126738, 8.820348729487620508323119462324