L(s) = 1 | + (0.720 + 2.21i)3-s + (−2.02 − 0.951i)5-s − 3.77·7-s + (−1.97 + 1.43i)9-s + (−3.05 − 2.21i)11-s + (−2.56 + 1.86i)13-s + (0.651 − 5.17i)15-s + (−0.430 + 1.32i)17-s + (−1.20 + 3.72i)19-s + (−2.72 − 8.37i)21-s + (0.720 + 0.523i)23-s + (3.19 + 3.84i)25-s + (1.05 + 0.765i)27-s + (0.0152 + 0.0468i)29-s + (1.72 − 5.30i)31-s + ⋯ |
L(s) = 1 | + (0.416 + 1.28i)3-s + (−0.905 − 0.425i)5-s − 1.42·7-s + (−0.658 + 0.478i)9-s + (−0.920 − 0.668i)11-s + (−0.712 + 0.517i)13-s + (0.168 − 1.33i)15-s + (−0.104 + 0.321i)17-s + (−0.277 + 0.853i)19-s + (−0.593 − 1.82i)21-s + (0.150 + 0.109i)23-s + (0.638 + 0.769i)25-s + (0.202 + 0.147i)27-s + (0.00282 + 0.00869i)29-s + (0.309 − 0.953i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.249i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 + 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0420238 - 0.331316i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0420238 - 0.331316i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (2.02 + 0.951i)T \) |
good | 3 | \( 1 + (-0.720 - 2.21i)T + (-2.42 + 1.76i)T^{2} \) |
| 7 | \( 1 + 3.77T + 7T^{2} \) |
| 11 | \( 1 + (3.05 + 2.21i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (2.56 - 1.86i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (0.430 - 1.32i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (1.20 - 3.72i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-0.720 - 0.523i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.0152 - 0.0468i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-1.72 + 5.30i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (5.70 - 4.14i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-1.20 + 0.875i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 2.69T + 43T^{2} \) |
| 47 | \( 1 + (1.16 + 3.58i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-3.58 - 11.0i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (0.558 - 0.405i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (8.38 + 6.08i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (4.73 - 14.5i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (-2.06 - 6.36i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (4.18 + 3.03i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-0.558 - 1.71i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-3.08 + 9.47i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (11.7 + 8.52i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (0.0278 + 0.0857i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−11.72049052095637057617676132839, −10.55479267670340013209211453656, −9.978467321056213188126123493221, −9.104964994170570242178531925774, −8.344854206550480789618168703697, −7.21724390227484249979193346712, −5.89393084687250488805917789463, −4.67902000246703755169721967939, −3.76304272698425360934395577337, −2.95219265609610371306191918141,
0.19352565241909321880128901247, 2.46888177894029131907025561502, 3.24034181109314724638537390476, 4.87895634485615564984923473798, 6.47543168819648083765533544259, 7.12097215209946343573522711810, 7.70167288348818423457499210911, 8.746405602530225815430706756850, 9.923153870730867936347195681521, 10.76070744903176587287894596777