L(s) = 1 | − 2.89i·3-s − i·5-s − 2.09i·7-s − 5.39·9-s + 4.99i·11-s + 6.59·13-s − 2.89·15-s − 1.38·19-s − 6.06·21-s + 6.69i·23-s − 25-s + 6.94i·27-s − 1.41i·29-s + 8.89i·31-s + 14.4·33-s + ⋯ |
L(s) = 1 | − 1.67i·3-s − 0.447i·5-s − 0.791i·7-s − 1.79·9-s + 1.50i·11-s + 1.82·13-s − 0.748·15-s − 0.317·19-s − 1.32·21-s + 1.39i·23-s − 0.200·25-s + 1.33i·27-s − 0.263i·29-s + 1.59i·31-s + 2.52·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.669 + 0.743i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.669 + 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.956902255\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.956902255\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 17 | \( 1 \) |
good | 3 | \( 1 + 2.89iT - 3T^{2} \) |
| 7 | \( 1 + 2.09iT - 7T^{2} \) |
| 11 | \( 1 - 4.99iT - 11T^{2} \) |
| 13 | \( 1 - 6.59T + 13T^{2} \) |
| 19 | \( 1 + 1.38T + 19T^{2} \) |
| 23 | \( 1 - 6.69iT - 23T^{2} \) |
| 29 | \( 1 + 1.41iT - 29T^{2} \) |
| 31 | \( 1 - 8.89iT - 31T^{2} \) |
| 37 | \( 1 - 11.2iT - 37T^{2} \) |
| 41 | \( 1 + 3.58iT - 41T^{2} \) |
| 43 | \( 1 - 9.22T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 + 0.738T + 53T^{2} \) |
| 59 | \( 1 + 2.25T + 59T^{2} \) |
| 61 | \( 1 - 2.60iT - 61T^{2} \) |
| 67 | \( 1 + 4.93T + 67T^{2} \) |
| 71 | \( 1 + 2.28iT - 71T^{2} \) |
| 73 | \( 1 - 4.04iT - 73T^{2} \) |
| 79 | \( 1 - 9.81iT - 79T^{2} \) |
| 83 | \( 1 - 11.6T + 83T^{2} \) |
| 89 | \( 1 - 3.27T + 89T^{2} \) |
| 97 | \( 1 - 4.32iT - 97T^{2} \) |
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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
−7.77686046663388406952542933611, −7.35116708585191607784319866451, −6.72767046699314317406419150044, −6.10898106874648822037951299912, −5.30720653079878296063079879454, −4.30213463673665493407444687391, −3.53944223180996149347786564857, −2.37456004280306649540959467881, −1.38662474364717716826621700149, −1.09265821673766582489732447262,
0.59259457427610731758400153006, 2.37536895005721489909187332891, 3.10668690957504652328172756506, 3.89696230560168303645667385430, 4.27268796559086347049391819378, 5.53759262264223874416429241032, 5.87921637088863441394264849646, 6.36952437032563394594642843735, 7.76260394825758697750199315811, 8.528586407175600881769349641699