L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.0880 − 0.0508i)3-s + (−0.499 − 0.866i)4-s + (−1.69 + 1.46i)5-s + 0.101i·6-s + (1.94 − 1.12i)7-s + 0.999·8-s + (−1.49 + 2.58i)9-s + (−0.419 − 2.19i)10-s − 3.35·11-s + (−0.0880 − 0.0508i)12-s + (2.04 + 3.54i)13-s + 2.25i·14-s + (−0.0747 + 0.214i)15-s + (−0.5 + 0.866i)16-s + (−0.819 + 1.41i)17-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.0508 − 0.0293i)3-s + (−0.249 − 0.433i)4-s + (−0.756 + 0.653i)5-s + 0.0414i·6-s + (0.736 − 0.425i)7-s + 0.353·8-s + (−0.498 + 0.863i)9-s + (−0.132 − 0.694i)10-s − 1.01·11-s + (−0.0254 − 0.0146i)12-s + (0.568 + 0.984i)13-s + 0.601i·14-s + (−0.0192 + 0.0554i)15-s + (−0.125 + 0.216i)16-s + (−0.198 + 0.344i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.446i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.151639 + 0.643004i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.151639 + 0.643004i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 + (1.69 - 1.46i)T \) |
| 37 | \( 1 + (-3.87 - 4.69i)T \) |
good | 3 | \( 1 + (-0.0880 + 0.0508i)T + (1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + (-1.94 + 1.12i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + 3.35T + 11T^{2} \) |
| 13 | \( 1 + (-2.04 - 3.54i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (0.819 - 1.41i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4.13 - 2.38i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 5.63T + 23T^{2} \) |
| 29 | \( 1 - 7.65iT - 29T^{2} \) |
| 31 | \( 1 + 1.11iT - 31T^{2} \) |
| 41 | \( 1 + (2.65 + 4.59i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 - 3.28T + 43T^{2} \) |
| 47 | \( 1 + 3.58iT - 47T^{2} \) |
| 53 | \( 1 + (0.873 + 0.504i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.82 - 1.63i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.44 + 2.56i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-12.9 + 7.48i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.50 - 2.61i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 9.63iT - 73T^{2} \) |
| 79 | \( 1 + (3.11 - 1.80i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-6.97 - 4.02i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-4.15 - 2.40i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 8.86T + 97T^{2} \) |
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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
−11.41247651897687010143245096039, −10.85370019314151606944477972093, −10.15297405969180063039394082434, −8.527417080740982403625544768735, −8.118519515419519710867906359643, −7.25753175043730799721373097660, −6.21296715585374775601380423930, −4.95422961870055290315573720877, −3.89799300023056099354579885790, −2.11191662048776264760291747434,
0.48570147145681414295982971779, 2.44157610261718983686203392305, 3.77699112772650151080570599084, 4.89547940239739148754120889769, 6.03498186356067052525040354154, 7.76168575261113498170843724678, 8.279532676223374970817955853235, 9.029080923405987947607637738110, 10.16219958301750084756791996694, 11.21389353730987394083607306616