| L(s) = 1 | + (−1.25 + 0.652i)2-s + (0.471 + 0.881i)3-s + (1.14 − 1.63i)4-s + (−0.827 + 0.678i)5-s + (−1.16 − 0.799i)6-s + (3.64 − 2.43i)7-s + (−0.375 + 2.80i)8-s + (−0.555 + 0.831i)9-s + (0.595 − 1.39i)10-s + (−0.341 − 1.12i)11-s + (1.98 + 0.242i)12-s + (1.25 − 1.52i)13-s + (−2.98 + 5.42i)14-s + (−0.988 − 0.409i)15-s + (−1.35 − 3.76i)16-s + (5.01 − 2.07i)17-s + ⋯ |
| L(s) = 1 | + (−0.887 + 0.461i)2-s + (0.272 + 0.509i)3-s + (0.574 − 0.818i)4-s + (−0.369 + 0.303i)5-s + (−0.476 − 0.326i)6-s + (1.37 − 0.919i)7-s + (−0.132 + 0.991i)8-s + (−0.185 + 0.277i)9-s + (0.188 − 0.439i)10-s + (−0.102 − 0.339i)11-s + (0.573 + 0.0699i)12-s + (0.347 − 0.423i)13-s + (−0.797 + 1.45i)14-s + (−0.255 − 0.105i)15-s + (−0.339 − 0.940i)16-s + (1.21 − 0.504i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.849 - 0.528i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.849 - 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.06035 + 0.302915i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.06035 + 0.302915i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.25 - 0.652i)T \) |
| 3 | \( 1 + (-0.471 - 0.881i)T \) |
| good | 5 | \( 1 + (0.827 - 0.678i)T + (0.975 - 4.90i)T^{2} \) |
| 7 | \( 1 + (-3.64 + 2.43i)T + (2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (0.341 + 1.12i)T + (-9.14 + 6.11i)T^{2} \) |
| 13 | \( 1 + (-1.25 + 1.52i)T + (-2.53 - 12.7i)T^{2} \) |
| 17 | \( 1 + (-5.01 + 2.07i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (0.652 - 0.0643i)T + (18.6 - 3.70i)T^{2} \) |
| 23 | \( 1 + (-3.05 + 0.608i)T + (21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (2.73 + 0.829i)T + (24.1 + 16.1i)T^{2} \) |
| 31 | \( 1 + (-5.87 - 5.87i)T + 31iT^{2} \) |
| 37 | \( 1 + (-0.224 + 2.27i)T + (-36.2 - 7.21i)T^{2} \) |
| 41 | \( 1 + (0.657 + 3.30i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (4.47 - 8.38i)T + (-23.8 - 35.7i)T^{2} \) |
| 47 | \( 1 + (-3.21 - 7.75i)T + (-33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (1.49 - 0.453i)T + (44.0 - 29.4i)T^{2} \) |
| 59 | \( 1 + (-0.765 - 0.932i)T + (-11.5 + 57.8i)T^{2} \) |
| 61 | \( 1 + (-7.45 + 3.98i)T + (33.8 - 50.7i)T^{2} \) |
| 67 | \( 1 + (2.70 - 1.44i)T + (37.2 - 55.7i)T^{2} \) |
| 71 | \( 1 + (-2.92 - 4.38i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (10.7 + 7.15i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (-2.74 + 6.61i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (1.51 + 15.4i)T + (-81.4 + 16.1i)T^{2} \) |
| 89 | \( 1 + (2.82 + 0.562i)T + (82.2 + 34.0i)T^{2} \) |
| 97 | \( 1 + (9.84 + 9.84i)T + 97iT^{2} \) |
| 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
−11.01973531293936391425091638747, −10.56513823147977249411115066067, −9.586985880684439545581620522999, −8.447082851354014605815086766296, −7.83405373549947685048696562069, −7.10969457534445273330017767686, −5.62158325444699409639768916236, −4.64654140105475878965639067587, −3.15598870623794595340311565173, −1.24644197927274465222108946019,
1.37657937882190604382777638319, 2.47903709152467694695322531645, 4.03961958080419643679498369106, 5.47651147243591998247570835017, 6.85130748835006540091477772937, 8.095127400696118435899545863302, 8.233125207157218546855267169379, 9.254343752273948088049881119459, 10.33738182641652218432892884978, 11.48102649058033867873949442747
