| L(s) = 1 | − 1.73i·3-s + 6.29·5-s + 2.64i·7-s − 2.99·9-s − 16.5i·11-s − 14.1·13-s − 10.8i·15-s − 20.4·17-s − 6.39i·19-s + 4.58·21-s + 29.0i·23-s + 14.5·25-s + 5.19i·27-s − 54.6·29-s − 14.7i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 1.25·5-s + 0.377i·7-s − 0.333·9-s − 1.50i·11-s − 1.08·13-s − 0.726i·15-s − 1.20·17-s − 0.336i·19-s + 0.218·21-s + 1.26i·23-s + 0.583·25-s + 0.192i·27-s − 1.88·29-s − 0.475i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6972980165\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6972980165\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| 7 | \( 1 - 2.64iT \) |
| good | 5 | \( 1 - 6.29T + 25T^{2} \) |
| 11 | \( 1 + 16.5iT - 121T^{2} \) |
| 13 | \( 1 + 14.1T + 169T^{2} \) |
| 17 | \( 1 + 20.4T + 289T^{2} \) |
| 19 | \( 1 + 6.39iT - 361T^{2} \) |
| 23 | \( 1 - 29.0iT - 529T^{2} \) |
| 29 | \( 1 + 54.6T + 841T^{2} \) |
| 31 | \( 1 + 14.7iT - 961T^{2} \) |
| 37 | \( 1 + 26.8T + 1.36e3T^{2} \) |
| 41 | \( 1 + 33.5T + 1.68e3T^{2} \) |
| 43 | \( 1 + 3.83iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 27.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 21.7T + 2.80e3T^{2} \) |
| 59 | \( 1 - 54.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 35.7T + 3.72e3T^{2} \) |
| 67 | \( 1 + 95.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 55.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 130.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 15.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 49.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 39.2T + 7.92e3T^{2} \) |
| 97 | \( 1 + 171.T + 9.40e3T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.166990565096219587258027584737, −8.283190687872264232595832780215, −7.29031376756199167462765353891, −6.46871806662027178777662270397, −5.66588382451877625305889519182, −5.18101182989554213760701990290, −3.59592524127049180421950429163, −2.46463181534381209958310861604, −1.74326019648587161615052525785, −0.17113987515956676400818955415,
1.85004898980956121352778875709, 2.47488133900435770534824268457, 3.99764164683947346150407015579, 4.81493957910421931686239927879, 5.48693742897212465494284078196, 6.65683517283155025328272498895, 7.16072853250710587303704395880, 8.360304089005225183439381203463, 9.385883184492363308586961131566, 9.743018613429851354996227892334
