L(s) = 1 | + (1.5 + 0.866i)2-s + 1.73i·3-s + (0.5 + 0.866i)4-s + (0.5 − 0.866i)5-s + (−1.49 + 2.59i)6-s + (−2.5 − 0.866i)7-s − 1.73i·8-s − 2.99·9-s + (1.5 − 0.866i)10-s + (3 − 1.73i)11-s + (−1.50 + 0.866i)12-s + 3.46i·13-s + (−3 − 3.46i)14-s + (1.49 + 0.866i)15-s + (2.49 − 4.33i)16-s + (3 + 5.19i)17-s + ⋯ |
L(s) = 1 | + (1.06 + 0.612i)2-s + 0.999i·3-s + (0.250 + 0.433i)4-s + (0.223 − 0.387i)5-s + (−0.612 + 1.06i)6-s + (−0.944 − 0.327i)7-s − 0.612i·8-s − 0.999·9-s + (0.474 − 0.273i)10-s + (0.904 − 0.522i)11-s + (−0.433 + 0.249i)12-s + 0.960i·13-s + (−0.801 − 0.925i)14-s + (0.387 + 0.223i)15-s + (0.624 − 1.08i)16-s + (0.727 + 1.26i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.31693 + 0.817032i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.31693 + 0.817032i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 1.73iT \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (2.5 + 0.866i)T \) |
good | 2 | \( 1 + (-1.5 - 0.866i)T + (1 + 1.73i)T^{2} \) |
| 11 | \( 1 + (-3 + 1.73i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 3.46iT - 13T^{2} \) |
| 17 | \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (6 + 3.46i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.5 + 0.866i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 1.73iT - 29T^{2} \) |
| 31 | \( 1 + (3 - 1.73i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2 - 3.46i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 - T + 43T^{2} \) |
| 47 | \( 1 + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4.5 + 2.59i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.5 - 11.2i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 6.92iT - 71T^{2} \) |
| 73 | \( 1 + (-3 + 1.73i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-8 + 13.8i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 9T + 83T^{2} \) |
| 89 | \( 1 + (-1.5 + 2.59i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 10.3iT - 97T^{2} \) |
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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
−14.16221337083112004545461272380, −13.16528479108188247367854058641, −12.17778866930277049334532663122, −10.71486310466920217838917030012, −9.638832843579580143076439947509, −8.713229594752641254721866506392, −6.64618256560601271719208051527, −5.88733815386282041729798579625, −4.43470446750789562720769364405, −3.61524603998433737121217567054,
2.35185031018607232999486137825, 3.59483606009000309654685246745, 5.53579999814029133993583250022, 6.51566663884681990143475985179, 7.88175014359809323304505485828, 9.371376049804196766070883257826, 10.78827111660785066695676928056, 12.07714230436148351318471699109, 12.47457273420476726697474355545, 13.40234493524462221707469713664