L(s) = 1 | + (0.236 + 2.70i)2-s + (−1.48 − 0.899i)3-s + (−5.28 + 0.931i)4-s + (−2.06 + 0.964i)5-s + (2.08 − 4.21i)6-s + (1.79 + 0.653i)7-s + (−2.36 − 8.81i)8-s + (1.38 + 2.66i)9-s + (−3.09 − 5.36i)10-s + (−1.78 + 3.08i)11-s + (8.65 + 3.37i)12-s + (−1.71 + 2.45i)13-s + (−1.34 + 5.00i)14-s + (3.92 + 0.431i)15-s + (13.1 − 4.80i)16-s + (0.742 + 1.06i)17-s + ⋯ |
L(s) = 1 | + (0.167 + 1.91i)2-s + (−0.854 − 0.519i)3-s + (−2.64 + 0.465i)4-s + (−0.924 + 0.431i)5-s + (0.849 − 1.72i)6-s + (0.678 + 0.246i)7-s + (−0.835 − 3.11i)8-s + (0.461 + 0.887i)9-s + (−0.979 − 1.69i)10-s + (−0.537 + 0.930i)11-s + (2.49 + 0.972i)12-s + (−0.476 + 0.680i)13-s + (−0.358 + 1.33i)14-s + (1.01 + 0.111i)15-s + (3.29 − 1.20i)16-s + (0.180 + 0.257i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 + 0.443i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.896 + 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.122886 - 0.525859i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.122886 - 0.525859i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.48 + 0.899i)T \) |
| 37 | \( 1 + (1.22 - 5.95i)T \) |
good | 2 | \( 1 + (-0.236 - 2.70i)T + (-1.96 + 0.347i)T^{2} \) |
| 5 | \( 1 + (2.06 - 0.964i)T + (3.21 - 3.83i)T^{2} \) |
| 7 | \( 1 + (-1.79 - 0.653i)T + (5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (1.78 - 3.08i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.71 - 2.45i)T + (-4.44 - 12.2i)T^{2} \) |
| 17 | \( 1 + (-0.742 - 1.06i)T + (-5.81 + 15.9i)T^{2} \) |
| 19 | \( 1 + (-1.14 - 0.0998i)T + (18.7 + 3.29i)T^{2} \) |
| 23 | \( 1 + (-4.25 - 1.13i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (2.33 - 0.625i)T + (25.1 - 14.5i)T^{2} \) |
| 31 | \( 1 + (-5.97 + 5.97i)T - 31iT^{2} \) |
| 41 | \( 1 + (0.817 + 4.63i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-6.64 - 6.64i)T + 43iT^{2} \) |
| 47 | \( 1 + (3.72 - 2.14i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.962 + 2.64i)T + (-40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (-1.35 + 2.89i)T + (-37.9 - 45.1i)T^{2} \) |
| 61 | \( 1 + (1.60 + 1.12i)T + (20.8 + 57.3i)T^{2} \) |
| 67 | \( 1 + (0.904 - 2.48i)T + (-51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-4.68 - 5.58i)T + (-12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 - 11.5iT - 73T^{2} \) |
| 79 | \( 1 + (1.05 + 2.27i)T + (-50.7 + 60.5i)T^{2} \) |
| 83 | \( 1 + (-17.7 - 3.13i)T + (77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (2.51 + 1.17i)T + (57.2 + 68.1i)T^{2} \) |
| 97 | \( 1 + (-0.168 + 0.629i)T + (-84.0 - 48.5i)T^{2} \) |
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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
−14.62040206297053995451510151251, −13.42077467592864839737342125385, −12.41279177566347881703452766793, −11.36315968741725932221439935388, −9.723034267136615043830315352232, −8.109698915058157431165804740267, −7.47455471233258359484597619929, −6.65641935739723685212143902225, −5.29132669084825521786497312982, −4.40814573223538105941328271945,
0.67523311792734454372056610515, 3.26819385968632531044897519303, 4.52877600229561371733786229774, 5.33903660205189823352575302947, 8.000911171414938534727860153464, 9.169123763652822102733800895801, 10.46874570771811361799201870768, 10.99593593115111769015636963695, 11.88803017795850049172883435086, 12.50868715050334388443840421141