L(s) = 1 | + (0.956 − 1.04i)2-s + 3-s + (−0.168 − 1.99i)4-s + 1.36i·5-s + (0.956 − 1.04i)6-s + 1.56·7-s + (−2.23 − 1.73i)8-s + 9-s + (1.41 + 1.30i)10-s + 0.528·11-s + (−0.168 − 1.99i)12-s − 2.52i·13-s + (1.49 − 1.63i)14-s + 1.36i·15-s + (−3.94 + 0.671i)16-s + 5.97·17-s + ⋯ |
L(s) = 1 | + (0.676 − 0.736i)2-s + 0.577·3-s + (−0.0841 − 0.996i)4-s + 0.609i·5-s + (0.390 − 0.425i)6-s + 0.592·7-s + (−0.790 − 0.612i)8-s + 0.333·9-s + (0.448 + 0.412i)10-s + 0.159·11-s + (−0.0485 − 0.575i)12-s − 0.699i·13-s + (0.400 − 0.435i)14-s + 0.351i·15-s + (−0.985 + 0.167i)16-s + 1.44·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 + 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.27978 - 1.66194i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.27978 - 1.66194i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.956 + 1.04i)T \) |
| 3 | \( 1 - T \) |
| 67 | \( 1 + (7.97 - 1.83i)T \) |
good | 5 | \( 1 - 1.36iT - 5T^{2} \) |
| 7 | \( 1 - 1.56T + 7T^{2} \) |
| 11 | \( 1 - 0.528T + 11T^{2} \) |
| 13 | \( 1 + 2.52iT - 13T^{2} \) |
| 17 | \( 1 - 5.97T + 17T^{2} \) |
| 19 | \( 1 + 2.51iT - 19T^{2} \) |
| 23 | \( 1 + 2.87iT - 23T^{2} \) |
| 29 | \( 1 + 7.65T + 29T^{2} \) |
| 31 | \( 1 - 8.15T + 31T^{2} \) |
| 37 | \( 1 + 7.80T + 37T^{2} \) |
| 41 | \( 1 - 11.2iT - 41T^{2} \) |
| 43 | \( 1 - 3.82T + 43T^{2} \) |
| 47 | \( 1 + 3.73iT - 47T^{2} \) |
| 53 | \( 1 - 11.2iT - 53T^{2} \) |
| 59 | \( 1 + 8.44iT - 59T^{2} \) |
| 61 | \( 1 + 2.04iT - 61T^{2} \) |
| 71 | \( 1 - 4.62iT - 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 + 2.66T + 79T^{2} \) |
| 83 | \( 1 - 11.3iT - 83T^{2} \) |
| 89 | \( 1 - 2.96T + 89T^{2} \) |
| 97 | \( 1 - 13.7iT - 97T^{2} \) |
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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
−10.25900660930030713564380573949, −9.493462434252992616910019033933, −8.445666742567241292804996120638, −7.52563296511294564135428971055, −6.49794671671763508667445669485, −5.44717331744737904505648151337, −4.52583461260720264710067595390, −3.35750533775337788420410056871, −2.67981943288559532368331824804, −1.28516962955804143452422108945,
1.69729307490600841819676987197, 3.24197726647804782595241139320, 4.15190984189346451393057826882, 5.09126081491252256776909650256, 5.88964647733137575902280175502, 7.13411602410049051261718000366, 7.77469858830957380364382343932, 8.604942534821345443805076624677, 9.228265766943792270229613219771, 10.31977795344550378985975445236