| L(s) = 1 | − 4.72·3-s + (−4.30 + 4.30i)5-s + (−7.52 − 7.52i)7-s + 13.3·9-s + (13.5 + 13.5i)11-s + (7.63 − 10.5i)13-s + (20.3 − 20.3i)15-s − 11.0i·17-s + (0.616 − 0.616i)19-s + (35.5 + 35.5i)21-s + 12.9i·23-s − 12.1i·25-s − 20.4·27-s + 10.1·29-s + (39.3 − 39.3i)31-s + ⋯ |
| L(s) = 1 | − 1.57·3-s + (−0.861 + 0.861i)5-s + (−1.07 − 1.07i)7-s + 1.48·9-s + (1.23 + 1.23i)11-s + (0.587 − 0.809i)13-s + (1.35 − 1.35i)15-s − 0.647i·17-s + (0.0324 − 0.0324i)19-s + (1.69 + 1.69i)21-s + 0.563i·23-s − 0.485i·25-s − 0.756·27-s + 0.351·29-s + (1.26 − 1.26i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.615203 - 0.103522i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.615203 - 0.103522i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + (-7.63 + 10.5i)T \) |
| good | 3 | \( 1 + 4.72T + 9T^{2} \) |
| 5 | \( 1 + (4.30 - 4.30i)T - 25iT^{2} \) |
| 7 | \( 1 + (7.52 + 7.52i)T + 49iT^{2} \) |
| 11 | \( 1 + (-13.5 - 13.5i)T + 121iT^{2} \) |
| 17 | \( 1 + 11.0iT - 289T^{2} \) |
| 19 | \( 1 + (-0.616 + 0.616i)T - 361iT^{2} \) |
| 23 | \( 1 - 12.9iT - 529T^{2} \) |
| 29 | \( 1 - 10.1T + 841T^{2} \) |
| 31 | \( 1 + (-39.3 + 39.3i)T - 961iT^{2} \) |
| 37 | \( 1 + (15.2 + 15.2i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 + (-10.7 + 10.7i)T - 1.68e3iT^{2} \) |
| 43 | \( 1 - 16.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (-3.09 - 3.09i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 - 7.27T + 2.80e3T^{2} \) |
| 59 | \( 1 + (26.1 + 26.1i)T + 3.48e3iT^{2} \) |
| 61 | \( 1 - 0.334T + 3.72e3T^{2} \) |
| 67 | \( 1 + (55.1 - 55.1i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 + (-68.7 + 68.7i)T - 5.04e3iT^{2} \) |
| 73 | \( 1 + (-55.5 - 55.5i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 - 65.3T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-64.8 + 64.8i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + (-70.7 - 70.7i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + (-120. + 120. i)T - 9.40e3iT^{2} \) |
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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
−11.90989407886861425666955706081, −11.21514156700972349173930942655, −10.35686771903929978410716037025, −9.616665550484125112139954519621, −7.53311784834646014419991403113, −6.86406676354450968900993737223, −6.13743729653147727085205353209, −4.51030300148089152985838601257, −3.52070883901656656591199160800, −0.65205382380756471470606044359,
0.889494487334164089952446533773, 3.65096594340045462682627227246, 4.87382175909109043794458198961, 6.21330535475831285403956225244, 6.45617226894547048787869569728, 8.475591281993314585030295357188, 9.103791594078141408108044408939, 10.52332102782393918908083161943, 11.61592725350327420635651171398, 12.03269227291547413878083900152
