| L(s) = 1 | + (1 − 1.73i)2-s + 1.73i·3-s + (−1.99 − 3.46i)4-s + (2.99 + 1.73i)6-s − 6.92i·7-s − 7.99·8-s − 2.99·9-s − 6.92i·11-s + (5.99 − 3.46i)12-s − 2·13-s + (−11.9 − 6.92i)14-s + (−8 + 13.8i)16-s − 10·17-s + (−2.99 + 5.19i)18-s − 20.7i·19-s + ⋯ |
| L(s) = 1 | + (0.5 − 0.866i)2-s + 0.577i·3-s + (−0.499 − 0.866i)4-s + (0.499 + 0.288i)6-s − 0.989i·7-s − 0.999·8-s − 0.333·9-s − 0.629i·11-s + (0.499 − 0.288i)12-s − 0.153·13-s + (−0.857 − 0.494i)14-s + (−0.5 + 0.866i)16-s − 0.588·17-s + (−0.166 + 0.288i)18-s − 1.09i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.499i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.866 + 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.369422 - 1.37870i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.369422 - 1.37870i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1 + 1.73i)T \) |
| 3 | \( 1 - 1.73iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 6.92iT - 49T^{2} \) |
| 11 | \( 1 + 6.92iT - 121T^{2} \) |
| 13 | \( 1 + 2T + 169T^{2} \) |
| 17 | \( 1 + 10T + 289T^{2} \) |
| 19 | \( 1 + 20.7iT - 361T^{2} \) |
| 23 | \( 1 + 27.7iT - 529T^{2} \) |
| 29 | \( 1 + 26T + 841T^{2} \) |
| 31 | \( 1 + 6.92iT - 961T^{2} \) |
| 37 | \( 1 + 26T + 1.36e3T^{2} \) |
| 41 | \( 1 - 58T + 1.68e3T^{2} \) |
| 43 | \( 1 + 48.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 69.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 74T + 2.80e3T^{2} \) |
| 59 | \( 1 - 90.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 26T + 3.72e3T^{2} \) |
| 67 | \( 1 - 6.92iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 46T + 5.32e3T^{2} \) |
| 79 | \( 1 + 117. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 48.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 82T + 7.92e3T^{2} \) |
| 97 | \( 1 + 2T + 9.40e3T^{2} \) |
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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
−10.86914994374009805816159690988, −10.62299954526491796619537476218, −9.435471181361320924532763970298, −8.649089061133141673114294565510, −7.09945477237034389091704740299, −5.85771213968725095321811117243, −4.64470667236947949773624715748, −3.86618194896178141329954625893, −2.56556922794828803380517569702, −0.57865852328494444581135371352,
2.19432048161390555066076840696, 3.71325821038048273749213094508, 5.17563019302033378856878539143, 5.95593084534368940392427728760, 7.02264388070960045120396929350, 7.88570578011239577205143688142, 8.830925477234147926690657007488, 9.721965955461200937777429923581, 11.37318921934460080913605010240, 12.20676232316273261844479902413
