L(s) = 1 | − 1.73i·7-s − 13-s − 1.73i·19-s + 1.73i·31-s − 2·37-s − 1.73i·43-s − 1.99·49-s − 61-s − 1.73i·67-s + 2·73-s + 1.73i·91-s − 97-s + 109-s + ⋯ |
L(s) = 1 | − 1.73i·7-s − 13-s − 1.73i·19-s + 1.73i·31-s − 2·37-s − 1.73i·43-s − 1.99·49-s − 61-s − 1.73i·67-s + 2·73-s + 1.73i·91-s − 97-s + 109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9009694597\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9009694597\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 1.73iT - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + 1.73iT - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 - 1.73iT - T^{2} \) |
| 37 | \( 1 + 2T + T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + 1.73iT - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 + 1.73iT - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 2T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + T + T^{2} \) |
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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
−8.495906941142676337876323612065, −7.51034146867927298879897266666, −7.02991515684197995930810338388, −6.62151177421089918367420348919, −5.14818961518955227816194293011, −4.79065477133777642667361615446, −3.80687696692475507210282410059, −3.04733404124779388086795473369, −1.80936154711806098453555856754, −0.49691925903654501326642723863,
1.76418639871973891381928831482, 2.49080917123977941079809505530, 3.40572694725035332119634689387, 4.48193437396371786389006378033, 5.40928291195063443796998300218, 5.83527989471391811188545608707, 6.64105846156008902713675095987, 7.70009629641350050643809782021, 8.211667345403131720149641876132, 8.980017735857843812538456317398