L(s) = 1 | + 29.2·2-s + (12.3 − 80.0i)3-s + 596.·4-s + (−218. + 585. i)5-s + (360. − 2.33e3i)6-s − 2.07e3i·7-s + 9.94e3·8-s + (−6.25e3 − 1.97e3i)9-s + (−6.36e3 + 1.71e4i)10-s + 1.19e4i·11-s + (7.37e3 − 4.77e4i)12-s + 2.78e4i·13-s − 6.05e4i·14-s + (4.41e4 + 2.46e4i)15-s + 1.37e5·16-s − 3.24e4·17-s + ⋯ |
L(s) = 1 | + 1.82·2-s + (0.152 − 0.988i)3-s + 2.33·4-s + (−0.348 + 0.937i)5-s + (0.278 − 1.80i)6-s − 0.863i·7-s + 2.42·8-s + (−0.953 − 0.301i)9-s + (−0.636 + 1.71i)10-s + 0.817i·11-s + (0.355 − 2.30i)12-s + 0.973i·13-s − 1.57i·14-s + (0.872 + 0.487i)15-s + 2.10·16-s − 0.388·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.487i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.872 + 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(3.96202 - 1.03185i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.96202 - 1.03185i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-12.3 + 80.0i)T \) |
| 5 | \( 1 + (218. - 585. i)T \) |
good | 2 | \( 1 - 29.2T + 256T^{2} \) |
| 7 | \( 1 + 2.07e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 - 1.19e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 2.78e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + 3.24e4T + 6.97e9T^{2} \) |
| 19 | \( 1 + 1.22e5T + 1.69e10T^{2} \) |
| 23 | \( 1 - 2.66e5T + 7.83e10T^{2} \) |
| 29 | \( 1 + 7.93e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 8.76e5T + 8.52e11T^{2} \) |
| 37 | \( 1 + 1.82e6iT - 3.51e12T^{2} \) |
| 41 | \( 1 - 1.49e5iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 1.65e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 - 3.35e6T + 2.38e13T^{2} \) |
| 53 | \( 1 - 2.01e6T + 6.22e13T^{2} \) |
| 59 | \( 1 + 2.16e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 1.66e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + 2.51e7iT - 4.06e14T^{2} \) |
| 71 | \( 1 - 4.99e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 4.82e6iT - 8.06e14T^{2} \) |
| 79 | \( 1 - 2.56e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + 4.83e7T + 2.25e15T^{2} \) |
| 89 | \( 1 + 3.21e6iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 5.09e7iT - 7.83e15T^{2} \) |
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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
−17.17613847933229865297683729145, −15.29730430071536893537271155320, −14.28549283805586308932214374577, −13.39574793912177312854112207513, −12.09999788992929339912223922137, −10.94304554288315515431392658957, −7.30980469891795708189353931008, −6.48409430132778583794303874794, −4.13100651422825263569974046180, −2.34073679726479718707650695803,
3.05001892709558637495892501345, 4.65841713091606243147240693911, 5.76289995489663972439963557638, 8.601882646810337980274564941907, 10.94474151575436244679812871513, 12.23361771051245334752027916780, 13.43128091508451968944500634219, 14.99629526478768962333650591861, 15.64617802445224942715793801585, 16.77193029837069411701389654602