L(s) = 1 | + i·2-s + (−0.950 + 1.44i)3-s − 4-s + 0.575·5-s + (−1.44 − 0.950i)6-s + (2.10 + 1.60i)7-s − i·8-s + (−1.19 − 2.75i)9-s + 0.575i·10-s − 6.32i·11-s + (0.950 − 1.44i)12-s − 2.57i·13-s + (−1.60 + 2.10i)14-s + (−0.547 + 0.834i)15-s + 16-s + 6.33·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (−0.548 + 0.836i)3-s − 0.5·4-s + 0.257·5-s + (−0.591 − 0.387i)6-s + (0.793 + 0.608i)7-s − 0.353i·8-s + (−0.398 − 0.917i)9-s + 0.182i·10-s − 1.90i·11-s + (0.274 − 0.418i)12-s − 0.714i·13-s + (−0.429 + 0.561i)14-s + (−0.141 + 0.215i)15-s + 0.250·16-s + 1.53·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 - 0.330i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 - 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.26270 + 0.214484i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.26270 + 0.214484i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + (0.950 - 1.44i)T \) |
| 7 | \( 1 + (-2.10 - 1.60i)T \) |
| 23 | \( 1 + iT \) |
good | 5 | \( 1 - 0.575T + 5T^{2} \) |
| 11 | \( 1 + 6.32iT - 11T^{2} \) |
| 13 | \( 1 + 2.57iT - 13T^{2} \) |
| 17 | \( 1 - 6.33T + 17T^{2} \) |
| 19 | \( 1 + 4.31iT - 19T^{2} \) |
| 29 | \( 1 + 10.1iT - 29T^{2} \) |
| 31 | \( 1 + 1.82iT - 31T^{2} \) |
| 37 | \( 1 + 3.23T + 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 - 6.83T + 43T^{2} \) |
| 47 | \( 1 - 1.58T + 47T^{2} \) |
| 53 | \( 1 - 4.99iT - 53T^{2} \) |
| 59 | \( 1 + 2.62T + 59T^{2} \) |
| 61 | \( 1 - 6.43iT - 61T^{2} \) |
| 67 | \( 1 + 1.55T + 67T^{2} \) |
| 71 | \( 1 + 4.59iT - 71T^{2} \) |
| 73 | \( 1 - 13.0iT - 73T^{2} \) |
| 79 | \( 1 - 16.7T + 79T^{2} \) |
| 83 | \( 1 - 2.97T + 83T^{2} \) |
| 89 | \( 1 - 3.91T + 89T^{2} \) |
| 97 | \( 1 + 17.9iT - 97T^{2} \) |
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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
−10.01150054797421824362719764117, −9.148516962939841462839571472573, −8.384540591547093233386575550009, −7.74579961405638034414227269717, −6.22963748987407420006070251447, −5.69368723143553897526341871259, −5.17970209534422547897833391828, −3.95549573411333806922356335505, −2.91547399551081518374988435689, −0.69853361620532613458978728612,
1.48354020946713119060383841713, 1.90400759339505292794490171290, 3.62354359477210596109103871198, 4.80360171439093749345022613751, 5.40339704754178308133316788902, 6.73715830969200436368176843552, 7.47801967629674234296139399572, 8.108913371434217547033370102000, 9.377250291113432554245956149081, 10.21987633219916308076244742380