L(s) = 1 | + 0.202·2-s − 2.69·3-s − 1.95·4-s − 0.545·6-s − 2.81·7-s − 0.800·8-s + 4.26·9-s − 1.84·11-s + 5.28·12-s − 6.49·13-s − 0.569·14-s + 3.75·16-s + 7.06·17-s + 0.863·18-s − 0.252·19-s + 7.59·21-s − 0.373·22-s − 23-s + 2.15·24-s − 1.31·26-s − 3.42·27-s + 5.51·28-s + 4.12·29-s + 3.54·31-s + 2.36·32-s + 4.98·33-s + 1.42·34-s + ⋯ |
L(s) = 1 | + 0.142·2-s − 1.55·3-s − 0.979·4-s − 0.222·6-s − 1.06·7-s − 0.283·8-s + 1.42·9-s − 0.557·11-s + 1.52·12-s − 1.80·13-s − 0.152·14-s + 0.939·16-s + 1.71·17-s + 0.203·18-s − 0.0578·19-s + 1.65·21-s − 0.0796·22-s − 0.208·23-s + 0.440·24-s − 0.257·26-s − 0.658·27-s + 1.04·28-s + 0.765·29-s + 0.637·31-s + 0.417·32-s + 0.867·33-s + 0.245·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3997122450\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3997122450\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 2 | \( 1 - 0.202T + 2T^{2} \) |
| 3 | \( 1 + 2.69T + 3T^{2} \) |
| 7 | \( 1 + 2.81T + 7T^{2} \) |
| 11 | \( 1 + 1.84T + 11T^{2} \) |
| 13 | \( 1 + 6.49T + 13T^{2} \) |
| 17 | \( 1 - 7.06T + 17T^{2} \) |
| 19 | \( 1 + 0.252T + 19T^{2} \) |
| 29 | \( 1 - 4.12T + 29T^{2} \) |
| 31 | \( 1 - 3.54T + 31T^{2} \) |
| 37 | \( 1 + 7.91T + 37T^{2} \) |
| 41 | \( 1 - 6.53T + 41T^{2} \) |
| 43 | \( 1 - 4.93T + 43T^{2} \) |
| 47 | \( 1 + 0.851T + 47T^{2} \) |
| 53 | \( 1 - 12.5T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 - 1.01T + 61T^{2} \) |
| 67 | \( 1 + 3.37T + 67T^{2} \) |
| 71 | \( 1 + 0.851T + 71T^{2} \) |
| 73 | \( 1 - 9.75T + 73T^{2} \) |
| 79 | \( 1 + 16.5T + 79T^{2} \) |
| 83 | \( 1 + 0.696T + 83T^{2} \) |
| 89 | \( 1 - 13.1T + 89T^{2} \) |
| 97 | \( 1 - 5.48T + 97T^{2} \) |
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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
−10.39813126754021039391436724793, −10.10902742575109900484095439112, −9.323899709994653495489664522610, −7.897663703571646391910324727414, −6.97468369245425863874344666852, −5.85414819978548281650352726865, −5.28206862910673912461070870857, −4.42693286368717701840507448025, −3.03493209632754417589126894702, −0.57214500208031481092954867687,
0.57214500208031481092954867687, 3.03493209632754417589126894702, 4.42693286368717701840507448025, 5.28206862910673912461070870857, 5.85414819978548281650352726865, 6.97468369245425863874344666852, 7.897663703571646391910324727414, 9.323899709994653495489664522610, 10.10902742575109900484095439112, 10.39813126754021039391436724793