L(s) = 1 | + 5.30i·3-s + (−4.75 + 1.54i)5-s + 0.206·7-s − 19.1·9-s − 15.0·11-s − 11.6·13-s + (−8.20 − 25.2i)15-s + 18.1i·17-s − 19.3·19-s + 1.09i·21-s + 27.2·23-s + (20.2 − 14.7i)25-s − 53.6i·27-s + 44.4i·29-s − 20.3i·31-s + ⋯ |
L(s) = 1 | + 1.76i·3-s + (−0.950 + 0.309i)5-s + 0.0295·7-s − 2.12·9-s − 1.36·11-s − 0.899·13-s + (−0.547 − 1.68i)15-s + 1.07i·17-s − 1.02·19-s + 0.0521i·21-s + 1.18·23-s + (0.808 − 0.588i)25-s − 1.98i·27-s + 1.53i·29-s − 0.657i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.891 + 0.453i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.891 + 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.07582080188\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.07582080188\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (4.75 - 1.54i)T \) |
good | 3 | \( 1 - 5.30iT - 9T^{2} \) |
| 7 | \( 1 - 0.206T + 49T^{2} \) |
| 11 | \( 1 + 15.0T + 121T^{2} \) |
| 13 | \( 1 + 11.6T + 169T^{2} \) |
| 17 | \( 1 - 18.1iT - 289T^{2} \) |
| 19 | \( 1 + 19.3T + 361T^{2} \) |
| 23 | \( 1 - 27.2T + 529T^{2} \) |
| 29 | \( 1 - 44.4iT - 841T^{2} \) |
| 31 | \( 1 + 20.3iT - 961T^{2} \) |
| 37 | \( 1 - 18.1T + 1.36e3T^{2} \) |
| 41 | \( 1 - 32.3T + 1.68e3T^{2} \) |
| 43 | \( 1 - 4.06iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 5.37T + 2.20e3T^{2} \) |
| 53 | \( 1 - 79.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + 83.3T + 3.48e3T^{2} \) |
| 61 | \( 1 + 36.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 4.51iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 41.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 41.5iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 15.5iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 50.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 10.8T + 7.92e3T^{2} \) |
| 97 | \( 1 - 12.1iT - 9.40e3T^{2} \) |
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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
−9.525664142111911694182818196619, −8.650550671427577052033560331062, −8.029926842163796025698981332824, −7.05845251513474711327043912360, −5.78220944300486672601706577295, −4.88522905638952787187368864933, −4.34691126194371572584429275957, −3.37422146522448551971967246030, −2.59731229183975550530958239957, −0.03014689718918358038164487782,
0.790508063346870422406089959621, 2.30376524404685983250621265921, 2.94393143741009669378018399596, 4.57529377480404553460913302650, 5.38724261151014847995045061788, 6.48968071780801638755191776098, 7.35067542442068270610090984030, 7.69434068203896062935270079231, 8.389193430093601420321276506800, 9.270063328742695618241321142867