L(s) = 1 | + 5.30·3-s + (1.54 + 4.75i)5-s + 0.206·7-s + 19.1·9-s − 15.0i·11-s − 11.6i·13-s + (8.20 + 25.2i)15-s + 18.1i·17-s + 19.3i·19-s + 1.09·21-s + 27.2·23-s + (−20.2 + 14.7i)25-s + 53.6·27-s − 44.4·29-s + 20.3i·31-s + ⋯ |
L(s) = 1 | + 1.76·3-s + (0.309 + 0.950i)5-s + 0.0295·7-s + 2.12·9-s − 1.36i·11-s − 0.899i·13-s + (0.547 + 1.68i)15-s + 1.07i·17-s + 1.02i·19-s + 0.0521·21-s + 1.18·23-s + (−0.808 + 0.588i)25-s + 1.98·27-s − 1.53·29-s + 0.657i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 - 0.309i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.950 - 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(3.06264 + 0.485964i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.06264 + 0.485964i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.54 - 4.75i)T \) |
good | 3 | \( 1 - 5.30T + 9T^{2} \) |
| 7 | \( 1 - 0.206T + 49T^{2} \) |
| 11 | \( 1 + 15.0iT - 121T^{2} \) |
| 13 | \( 1 + 11.6iT - 169T^{2} \) |
| 17 | \( 1 - 18.1iT - 289T^{2} \) |
| 19 | \( 1 - 19.3iT - 361T^{2} \) |
| 23 | \( 1 - 27.2T + 529T^{2} \) |
| 29 | \( 1 + 44.4T + 841T^{2} \) |
| 31 | \( 1 - 20.3iT - 961T^{2} \) |
| 37 | \( 1 + 18.1iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 32.3T + 1.68e3T^{2} \) |
| 43 | \( 1 + 4.06T + 1.84e3T^{2} \) |
| 47 | \( 1 + 5.37T + 2.20e3T^{2} \) |
| 53 | \( 1 + 79.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 83.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 36.7T + 3.72e3T^{2} \) |
| 67 | \( 1 + 4.51T + 4.48e3T^{2} \) |
| 71 | \( 1 - 41.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 41.5iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 15.5iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 50.9T + 6.88e3T^{2} \) |
| 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
−11.17603026171106609474254325784, −10.39670692823358065853404095612, −9.523614789384289176932882880307, −8.468604520831540677431471662776, −7.937908704103521943353026728950, −6.82760385975638745247936295757, −5.61313669076836199299669629477, −3.57944722071685217295176353003, −3.21716585131796481145108321573, −1.82805419541628823964891317981,
1.62945861158398125230322628435, 2.67461665552999545775035515485, 4.18612321409779024994200896705, 4.96994208557692158465404937959, 6.95122346041400800689815279691, 7.61160423374825013129483652523, 8.835337763442914526164736076382, 9.290547986247337122146133744876, 9.875891916254679581213591993353, 11.48843720903897682168210540532