L(s) = 1 | + 2·2-s + 2.23i·3-s − 2.23i·5-s + 4.47i·6-s + (−2 − 6.70i)7-s − 8·8-s + 3.99·9-s − 4.47i·10-s − 11-s + 20.1i·13-s + (−4 − 13.4i)14-s + 5.00·15-s − 16·16-s + 6.70i·17-s + 7.99·18-s − 13.4i·19-s + ⋯ |
L(s) = 1 | + 2-s + 0.745i·3-s − 0.447i·5-s + 0.745i·6-s + (−0.285 − 0.958i)7-s − 8-s + 0.444·9-s − 0.447i·10-s − 0.0909·11-s + 1.54i·13-s + (−0.285 − 0.958i)14-s + 0.333·15-s − 16-s + 0.394i·17-s + 0.444·18-s − 0.706i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.285i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.958 - 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.40821 + 0.205456i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.40821 + 0.205456i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + 2.23iT \) |
| 7 | \( 1 + (2 + 6.70i)T \) |
good | 2 | \( 1 - 2T + 4T^{2} \) |
| 3 | \( 1 - 2.23iT - 9T^{2} \) |
| 11 | \( 1 + T + 121T^{2} \) |
| 13 | \( 1 - 20.1iT - 169T^{2} \) |
| 17 | \( 1 - 6.70iT - 289T^{2} \) |
| 19 | \( 1 + 13.4iT - 361T^{2} \) |
| 23 | \( 1 - 8T + 529T^{2} \) |
| 29 | \( 1 - 41T + 841T^{2} \) |
| 31 | \( 1 + 40.2iT - 961T^{2} \) |
| 37 | \( 1 + 28T + 1.36e3T^{2} \) |
| 41 | \( 1 + 13.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 82T + 1.84e3T^{2} \) |
| 47 | \( 1 + 20.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 74T + 2.80e3T^{2} \) |
| 59 | \( 1 - 93.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 80.4iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 2T + 4.48e3T^{2} \) |
| 71 | \( 1 - 14T + 5.04e3T^{2} \) |
| 73 | \( 1 - 67.0iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 19T + 6.24e3T^{2} \) |
| 83 | \( 1 + 93.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 107. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 60.3iT - 9.40e3T^{2} \) |
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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
−16.19003335948102889435073738464, −15.10298044899458491757036242703, −13.86759591438522435402586277751, −13.08720305968395502503817355295, −11.70603651917400914192694043400, −10.11995477735037076388918724790, −8.957678794272703689224554903528, −6.75293607249556376196345518939, −4.79748046709348623827271120853, −3.91446018637379515645859447007,
3.05371396911243213951153324425, 5.27925486380411942242917969488, 6.61392551631443716343866201571, 8.355342312889916135652621542860, 10.11223005987859719346863047131, 12.03576960908349266341318005236, 12.71106252931408021522466085719, 13.70663786638314494099409636780, 14.95421579659256666946125634018, 15.78843731153310701546008509645