L(s) = 1 | + (1.73 + i)2-s + (−0.866 + 0.5i)3-s + (0.999 + 1.73i)4-s − 1.99·6-s + (2.59 − 0.5i)7-s + (0.499 − 0.866i)9-s + (3 + 5.19i)11-s + (−1.73 − i)12-s + 3i·13-s + (5 + 1.73i)14-s + (1.99 − 3.46i)16-s + (−3.46 + 2i)17-s + (1.73 − 0.999i)18-s + (0.5 − 0.866i)19-s + (−2 + 1.73i)21-s + 12i·22-s + ⋯ |
L(s) = 1 | + (1.22 + 0.707i)2-s + (−0.499 + 0.288i)3-s + (0.499 + 0.866i)4-s − 0.816·6-s + (0.981 − 0.188i)7-s + (0.166 − 0.288i)9-s + (0.904 + 1.56i)11-s + (−0.499 − 0.288i)12-s + 0.832i·13-s + (1.33 + 0.462i)14-s + (0.499 − 0.866i)16-s + (−0.840 + 0.485i)17-s + (0.408 − 0.235i)18-s + (0.114 − 0.198i)19-s + (−0.436 + 0.377i)21-s + 2.55i·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.208 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.208 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.98914 + 1.60961i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.98914 + 1.60961i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.866 - 0.5i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.59 + 0.5i)T \) |
good | 2 | \( 1 + (-1.73 - i)T + (1 + 1.73i)T^{2} \) |
| 11 | \( 1 + (-3 - 5.19i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 3iT - 13T^{2} \) |
| 17 | \( 1 + (3.46 - 2i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.46 + 2i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (6.06 + 3.5i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + iT - 43T^{2} \) |
| 47 | \( 1 + (1.73 + i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.46 - 2i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4 + 6.92i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-7 + 12.1i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.06 + 3.5i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + (0.866 - 0.5i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 2iT - 83T^{2} \) |
| 89 | \( 1 + (6 - 10.3i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 6iT - 97T^{2} \) |
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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
−11.32385797145800000184999378129, −10.24396884994689743155103379205, −9.345487884113468746679782644689, −8.120633472969096331888780512459, −6.82489351120759028108738776922, −6.61266813666980583828202704548, −5.18944761079774619361655332962, −4.51683677201848150417456923096, −3.95049740998953673640855107249, −1.86455872379287335964664445562,
1.33411197160609318755312097842, 2.78879427067648028942137348310, 3.91233971269841431758581248533, 4.98882105457079440425963807600, 5.70350134481724424894284273379, 6.61615645401728381683934064004, 8.102608414592475861344752028736, 8.709067426536040643826159491117, 10.31582802657754985728789211563, 11.06961328765490838852376873335