| L(s) = 1 | + (−1.45 − 1.37i)2-s + (2.65 + 1.39i)3-s + (0.232 + 3.99i)4-s + (1.27 + 7.23i)5-s + (−1.95 − 5.67i)6-s + (−3.68 − 3.08i)7-s + (5.14 − 6.12i)8-s + (5.13 + 7.39i)9-s + (8.07 − 12.2i)10-s + (1.42 − 8.09i)11-s + (−4.93 + 10.9i)12-s + (−7.79 + 21.4i)13-s + (1.11 + 9.54i)14-s + (−6.66 + 21.0i)15-s + (−15.8 + 1.86i)16-s + (−8.45 + 4.88i)17-s + ⋯ |
| L(s) = 1 | + (−0.727 − 0.686i)2-s + (0.886 + 0.463i)3-s + (0.0582 + 0.998i)4-s + (0.255 + 1.44i)5-s + (−0.326 − 0.945i)6-s + (−0.525 − 0.441i)7-s + (0.642 − 0.766i)8-s + (0.570 + 0.821i)9-s + (0.807 − 1.22i)10-s + (0.129 − 0.736i)11-s + (−0.411 + 0.911i)12-s + (−0.599 + 1.64i)13-s + (0.0797 + 0.681i)14-s + (−0.444 + 1.40i)15-s + (−0.993 + 0.116i)16-s + (−0.497 + 0.287i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.278 - 0.960i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.278 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.02203 + 0.768032i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.02203 + 0.768032i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.45 + 1.37i)T \) |
| 3 | \( 1 + (-2.65 - 1.39i)T \) |
| good | 5 | \( 1 + (-1.27 - 7.23i)T + (-23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (3.68 + 3.08i)T + (8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (-1.42 + 8.09i)T + (-113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (7.79 - 21.4i)T + (-129. - 108. i)T^{2} \) |
| 17 | \( 1 + (8.45 - 4.88i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (7.48 + 4.32i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-12.0 - 14.3i)T + (-91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (-43.9 + 15.9i)T + (644. - 540. i)T^{2} \) |
| 31 | \( 1 + (20.7 - 17.4i)T + (166. - 946. i)T^{2} \) |
| 37 | \( 1 + (32.8 - 18.9i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-6.01 + 16.5i)T + (-1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (-64.9 - 11.4i)T + (1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (-28.7 + 34.2i)T + (-383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 - 65.5T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-3.85 - 21.8i)T + (-3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (28.1 - 33.5i)T + (-646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-24.9 + 68.5i)T + (-3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (96.8 - 55.9i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-33.8 + 58.5i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-66.4 + 24.1i)T + (4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (-102. + 37.1i)T + (5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (46.3 + 26.7i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (21.8 - 124. i)T + (-8.84e3 - 3.21e3i)T^{2} \) |
| 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
−12.01046114505095665028039126780, −10.89903261264862636437882275691, −10.39018074442187722646247061740, −9.434475126656254927900057309985, −8.677373626487949184961630827834, −7.26179387733040699176367924967, −6.68266161903366781735209825605, −4.20197835700765999061936409567, −3.19465690481620194621052814355, −2.17933715853804750047842349254,
0.818667798248192593903953680664, 2.43452559352018920281157140766, 4.64504442803460151999207408045, 5.77483869597011612002766457659, 7.06656790119048473415861951724, 8.076597457356319748379296501995, 8.871542331479270605085209586895, 9.476743609199323552340293253767, 10.44994384913787038411631007085, 12.50900416325677176973394150265
