| L(s) = 1 | + (9.79 − 5.65i)2-s + (30.3 − 17.5i)3-s + (63.9 − 110. i)4-s + (−189. − 328. i)5-s + (198. − 342. i)6-s − 4.48e3·7-s − 1.44e3i·8-s + (−2.66e3 + 4.62e3i)9-s + (−3.71e3 − 2.14e3i)10-s + 1.08e4·11-s − 4.48e3i·12-s + (−2.22e4 − 1.28e4i)13-s + (−4.39e4 + 2.53e4i)14-s + (−1.14e4 − 6.63e3i)15-s + (−8.19e3 − 1.41e4i)16-s + (9.55e3 + 1.65e4i)17-s + ⋯ |
| L(s) = 1 | + (0.612 − 0.353i)2-s + (0.374 − 0.216i)3-s + (0.249 − 0.433i)4-s + (−0.303 − 0.525i)5-s + (0.152 − 0.264i)6-s − 1.86·7-s − 0.353i·8-s + (−0.406 + 0.704i)9-s + (−0.371 − 0.214i)10-s + 0.743·11-s − 0.216i·12-s + (−0.779 − 0.450i)13-s + (−1.14 + 0.660i)14-s + (−0.226 − 0.130i)15-s + (−0.125 − 0.216i)16-s + (0.114 + 0.198i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.154i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.987 - 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.0643412 + 0.826805i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0643412 + 0.826805i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-9.79 + 5.65i)T \) |
| 19 | \( 1 + (5.85e4 + 1.16e5i)T \) |
| good | 3 | \( 1 + (-30.3 + 17.5i)T + (3.28e3 - 5.68e3i)T^{2} \) |
| 5 | \( 1 + (189. + 328. i)T + (-1.95e5 + 3.38e5i)T^{2} \) |
| 7 | \( 1 + 4.48e3T + 5.76e6T^{2} \) |
| 11 | \( 1 - 1.08e4T + 2.14e8T^{2} \) |
| 13 | \( 1 + (2.22e4 + 1.28e4i)T + (4.07e8 + 7.06e8i)T^{2} \) |
| 17 | \( 1 + (-9.55e3 - 1.65e4i)T + (-3.48e9 + 6.04e9i)T^{2} \) |
| 23 | \( 1 + (1.70e5 - 2.95e5i)T + (-3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 + (6.38e5 + 3.68e5i)T + (2.50e11 + 4.33e11i)T^{2} \) |
| 31 | \( 1 + 6.58e5iT - 8.52e11T^{2} \) |
| 37 | \( 1 + 1.71e6iT - 3.51e12T^{2} \) |
| 41 | \( 1 + (6.68e5 - 3.85e5i)T + (3.99e12 - 6.91e12i)T^{2} \) |
| 43 | \( 1 + (8.28e5 + 1.43e6i)T + (-5.84e12 + 1.01e13i)T^{2} \) |
| 47 | \( 1 + (-3.48e6 + 6.03e6i)T + (-1.19e13 - 2.06e13i)T^{2} \) |
| 53 | \( 1 + (-7.32e6 - 4.23e6i)T + (3.11e13 + 5.39e13i)T^{2} \) |
| 59 | \( 1 + (-1.58e7 + 9.16e6i)T + (7.34e13 - 1.27e14i)T^{2} \) |
| 61 | \( 1 + (8.19e6 - 1.41e7i)T + (-9.58e13 - 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-1.33e7 - 7.70e6i)T + (2.03e14 + 3.51e14i)T^{2} \) |
| 71 | \( 1 + (3.74e7 - 2.16e7i)T + (3.22e14 - 5.59e14i)T^{2} \) |
| 73 | \( 1 + (2.50e7 + 4.34e7i)T + (-4.03e14 + 6.98e14i)T^{2} \) |
| 79 | \( 1 + (4.99e7 - 2.88e7i)T + (7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 - 1.97e7T + 2.25e15T^{2} \) |
| 89 | \( 1 + (-7.33e6 - 4.23e6i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 + (-9.58e6 + 5.53e6i)T + (3.91e15 - 6.78e15i)T^{2} \) |
| 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
−13.56230101217155785359672380848, −12.88542344605449510766336706768, −11.80408564034906155106790990593, −10.15228129632667053557447708857, −8.985823549989534550502169353036, −7.20538716989636059278257408185, −5.71277276577588418484776657397, −3.88435148081720848381125646370, −2.49473627687842288602663935881, −0.23809026283555761294054051914,
2.94371294096333421070121238520, 3.89454246629278186449168809561, 6.16259848924659311044783661611, 7.01463151512519875885178830233, 8.945715678688225672068339531227, 10.07626009530229819139706619567, 11.90010770624908603760160840643, 12.78361979502628097379139631856, 14.24804759585791071580685884906, 14.95495592238616506534638505281
