| L(s) = 1 | + (2.75 − 1.18i)3-s + 3.68·5-s + 5.43·7-s + (6.21 − 6.51i)9-s + 1.16·11-s + 14.4i·13-s + (10.1 − 4.35i)15-s − 1.71i·17-s − 28.8i·19-s + (14.9 − 6.42i)21-s + 35.4i·23-s − 11.4·25-s + (9.43 − 25.2i)27-s − 33.6·29-s + 55.5·31-s + ⋯ |
| L(s) = 1 | + (0.919 − 0.393i)3-s + 0.736·5-s + 0.776·7-s + (0.690 − 0.723i)9-s + 0.105·11-s + 1.10i·13-s + (0.677 − 0.290i)15-s − 0.100i·17-s − 1.51i·19-s + (0.714 − 0.305i)21-s + 1.54i·23-s − 0.456·25-s + (0.349 − 0.936i)27-s − 1.16·29-s + 1.79·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.371i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.928 + 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.78777 - 0.537343i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.78777 - 0.537343i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.75 + 1.18i)T \) |
| good | 5 | \( 1 - 3.68T + 25T^{2} \) |
| 7 | \( 1 - 5.43T + 49T^{2} \) |
| 11 | \( 1 - 1.16T + 121T^{2} \) |
| 13 | \( 1 - 14.4iT - 169T^{2} \) |
| 17 | \( 1 + 1.71iT - 289T^{2} \) |
| 19 | \( 1 + 28.8iT - 361T^{2} \) |
| 23 | \( 1 - 35.4iT - 529T^{2} \) |
| 29 | \( 1 + 33.6T + 841T^{2} \) |
| 31 | \( 1 - 55.5T + 961T^{2} \) |
| 37 | \( 1 + 30.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 63.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 43.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 61.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 56.3T + 2.80e3T^{2} \) |
| 59 | \( 1 - 49.6T + 3.48e3T^{2} \) |
| 61 | \( 1 - 4.73iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 7.08iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 96.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 68.5T + 5.32e3T^{2} \) |
| 79 | \( 1 + 9.72T + 6.24e3T^{2} \) |
| 83 | \( 1 + 38.9T + 6.88e3T^{2} \) |
| 89 | \( 1 + 0.675iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 86.1T + 9.40e3T^{2} \) |
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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
−11.24575296341829614475870259087, −9.784588395437011221323027946155, −9.288898899394205687771935672242, −8.344874722478289651747578588315, −7.34669979831401411596790798949, −6.49699039270020843555685391604, −5.15072117966152273714266910999, −3.95199203378341746415572621260, −2.46310221390575650289720942241, −1.48278573480424621464723944615,
1.61402519250403943970940273920, 2.81586264768879660838656765911, 4.13698746714490288867641004241, 5.23673642144779893326736174634, 6.33291508272230732609938646213, 7.916150188553928512073936876955, 8.206693516325961963369086790234, 9.439623307889156495060345518987, 10.18782107791675331674823937671, 10.85024646854570422333184653025
