| L(s) = 1 | − 1.73i·3-s − 2.23i·5-s + (−6.70 + 2.02i)7-s − 2.99·9-s − 11.9·11-s + 9.67i·13-s − 3.87·15-s − 7.09i·17-s − 17.5i·19-s + (3.50 + 11.6i)21-s + 2.84·23-s − 5.00·25-s + 5.19i·27-s − 13.6·29-s + 19.9i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.447i·5-s + (−0.957 + 0.288i)7-s − 0.333·9-s − 1.08·11-s + 0.744i·13-s − 0.258·15-s − 0.417i·17-s − 0.923i·19-s + (0.166 + 0.552i)21-s + 0.123·23-s − 0.200·25-s + 0.192i·27-s − 0.472·29-s + 0.643i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.078749062\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.078749062\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| 5 | \( 1 + 2.23iT \) |
| 7 | \( 1 + (6.70 - 2.02i)T \) |
| good | 11 | \( 1 + 11.9T + 121T^{2} \) |
| 13 | \( 1 - 9.67iT - 169T^{2} \) |
| 17 | \( 1 + 7.09iT - 289T^{2} \) |
| 19 | \( 1 + 17.5iT - 361T^{2} \) |
| 23 | \( 1 - 2.84T + 529T^{2} \) |
| 29 | \( 1 + 13.6T + 841T^{2} \) |
| 31 | \( 1 - 19.9iT - 961T^{2} \) |
| 37 | \( 1 + 11.0T + 1.36e3T^{2} \) |
| 41 | \( 1 + 28.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 72.9T + 1.84e3T^{2} \) |
| 47 | \( 1 - 28.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 11.7T + 2.80e3T^{2} \) |
| 59 | \( 1 - 101. iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 76.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 76.2T + 4.48e3T^{2} \) |
| 71 | \( 1 - 95.4T + 5.04e3T^{2} \) |
| 73 | \( 1 - 120. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 14.8T + 6.24e3T^{2} \) |
| 83 | \( 1 + 60.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 88.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 18.8iT - 9.40e3T^{2} \) |
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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
−9.092418605802626794350662230846, −8.554286437393892832803260581276, −7.42373763772242290666644393702, −6.96503026081041983306657017224, −5.94276908510597076412140090657, −5.26061897754695990539575073198, −4.22008270295211765722144075637, −2.97960740897888807289755722526, −2.23374757958894438647389030167, −0.75490505468220668939100917697,
0.40421529324936411724555952472, 2.27353865929681829889464268649, 3.26130884220568726608504505701, 3.87542116921018040274785763282, 5.10749067991496278180852542035, 5.86237454227588808919748695740, 6.62366002657525813342220236989, 7.71713552111055001956768903440, 8.174525080341201087028991614137, 9.441855748990509348121703532925
