L(s) = 1 | + (−0.268 − 0.465i)3-s + i·5-s + (0.331 + 0.191i)7-s + (1.35 − 2.34i)9-s + (−5.66 + 3.27i)11-s + (−2.44 + 2.65i)13-s + (0.465 − 0.268i)15-s + (0.174 − 0.301i)17-s + (−3.55 − 2.05i)19-s − 0.205i·21-s + (−3.36 − 5.82i)23-s − 25-s − 3.06·27-s + (1.91 + 3.32i)29-s − 1.67i·31-s + ⋯ |
L(s) = 1 | + (−0.155 − 0.268i)3-s + 0.447i·5-s + (0.125 + 0.0723i)7-s + (0.451 − 0.782i)9-s + (−1.70 + 0.986i)11-s + (−0.677 + 0.735i)13-s + (0.120 − 0.0693i)15-s + (0.0422 − 0.0732i)17-s + (−0.815 − 0.470i)19-s − 0.0449i·21-s + (−0.701 − 1.21i)23-s − 0.200·25-s − 0.590·27-s + (0.356 + 0.617i)29-s − 0.301i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 - 0.231i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.972 - 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1603377528\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1603377528\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 + (2.44 - 2.65i)T \) |
good | 3 | \( 1 + (0.268 + 0.465i)T + (-1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + (-0.331 - 0.191i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (5.66 - 3.27i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-0.174 + 0.301i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.55 + 2.05i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.36 + 5.82i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.91 - 3.32i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 1.67iT - 31T^{2} \) |
| 37 | \( 1 + (-0.963 + 0.556i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (6.17 - 3.56i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.87 - 4.98i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 7.72iT - 47T^{2} \) |
| 53 | \( 1 + 7.18T + 53T^{2} \) |
| 59 | \( 1 + (-3.85 - 2.22i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.19 + 9.00i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.01 - 1.74i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (2.93 + 1.69i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 8.04iT - 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 + 8.27iT - 83T^{2} \) |
| 89 | \( 1 + (15.5 - 8.97i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.14 - 0.661i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.12807562299318515487276487797, −9.770560697293753902409443914761, −8.568316864769988579161090149285, −7.69556419129544801815987986941, −6.92105056112210043529104464597, −6.29882834694920970256895470007, −4.99426503483521183657563845549, −4.33039256159540066188027444684, −2.85663224302850756463069232352, −1.95457308790070411542983616614,
0.06730994294964929536256026661, 1.94902640714682182589886758903, 3.13622909776747023176204562374, 4.37692448094327530816065889007, 5.32633759759153374689278501430, 5.73300162791103516397018860231, 7.23553825576206206437355148961, 8.056092974707672414227716452993, 8.418798509206239205229054526513, 9.845698766613489223287305175124