L(s) = 1 | + (1.02 − 0.590i)2-s + (1.63 + 0.577i)3-s + (−0.301 + 0.522i)4-s + (0.5 − 0.866i)5-s + (2.01 − 0.373i)6-s + (−2.07 + 1.63i)7-s + 3.07i·8-s + (2.33 + 1.88i)9-s − 1.18i·10-s + (5.38 − 3.10i)11-s + (−0.794 + 0.679i)12-s + (−0.534 − 0.308i)13-s + (−1.15 + 2.90i)14-s + (1.31 − 1.12i)15-s + (1.21 + 2.10i)16-s − 2.15·17-s + ⋯ |
L(s) = 1 | + (0.723 − 0.417i)2-s + (0.942 + 0.333i)3-s + (−0.150 + 0.261i)4-s + (0.223 − 0.387i)5-s + (0.821 − 0.152i)6-s + (−0.785 + 0.618i)7-s + 1.08i·8-s + (0.777 + 0.628i)9-s − 0.373i·10-s + (1.62 − 0.937i)11-s + (−0.229 + 0.196i)12-s + (−0.148 − 0.0855i)13-s + (−0.309 + 0.776i)14-s + (0.339 − 0.290i)15-s + (0.303 + 0.525i)16-s − 0.523·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.161i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 - 0.161i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.32886 + 0.188928i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.32886 + 0.188928i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.63 - 0.577i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (2.07 - 1.63i)T \) |
good | 2 | \( 1 + (-1.02 + 0.590i)T + (1 - 1.73i)T^{2} \) |
| 11 | \( 1 + (-5.38 + 3.10i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.534 + 0.308i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 2.15T + 17T^{2} \) |
| 19 | \( 1 + 4.75iT - 19T^{2} \) |
| 23 | \( 1 + (3.22 + 1.86i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (4.09 - 2.36i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (7.25 + 4.19i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 2.00T + 37T^{2} \) |
| 41 | \( 1 + (0.261 - 0.453i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.75 - 4.77i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.25 - 5.63i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 12.7iT - 53T^{2} \) |
| 59 | \( 1 + (2.72 - 4.72i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-12.3 + 7.13i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.146 - 0.254i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 5.45iT - 71T^{2} \) |
| 73 | \( 1 + 0.406iT - 73T^{2} \) |
| 79 | \( 1 + (2.85 + 4.94i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-6.40 - 11.1i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 0.957T + 89T^{2} \) |
| 97 | \( 1 + (2.17 - 1.25i)T + (48.5 - 84.0i)T^{2} \) |
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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
−11.83491032314463616591622940528, −10.98418604242406778837589699487, −9.385132169539822443045645755058, −9.090056375541302969509936882794, −8.208463846215916042876069899063, −6.73260476841475996308638392331, −5.47143792403948247780523003035, −4.18103572643380417339291316116, −3.41825279240569182754456956387, −2.24213303148084406885337585131,
1.71018965016651426456957325336, 3.65371928050415633119484928151, 4.14335587192123054203205981460, 5.91540085158503266474379953591, 6.84455672145950145888578331410, 7.34225474955013917939733837341, 9.043786903301672537243441534424, 9.640372600210915714382889848029, 10.40212553536793869815252641783, 12.08074789663775505780410096365