L(s) = 1 | + (1.5 + 0.866i)3-s + (−0.5 + 0.866i)7-s + (1 + 1.73i)9-s + (−1.5 + 0.866i)21-s + (−1.5 + 0.866i)23-s + 1.73i·27-s + 29-s + 41-s − 1.73i·43-s + (−0.499 − 0.866i)49-s + (−0.5 − 0.866i)61-s − 2·63-s + (1.5 + 0.866i)67-s − 3·69-s + (−0.5 + 0.866i)81-s + ⋯ |
L(s) = 1 | + (1.5 + 0.866i)3-s + (−0.5 + 0.866i)7-s + (1 + 1.73i)9-s + (−1.5 + 0.866i)21-s + (−1.5 + 0.866i)23-s + 1.73i·27-s + 29-s + 41-s − 1.73i·43-s + (−0.499 − 0.866i)49-s + (−0.5 − 0.866i)61-s − 2·63-s + (1.5 + 0.866i)67-s − 3·69-s + (−0.5 + 0.866i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.904220977\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.904220977\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
good | 3 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 - T + T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( 1 + 1.73iT - T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - 1.73iT - T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−9.156650924377871290781679716245, −8.490224826170751356798806729318, −7.996662931193688149785203249798, −7.07303648468975443445846057408, −6.01535145275547496174781841091, −5.19746664228429477241012078262, −4.17492027875520094134526216986, −3.53740111545661300441772818831, −2.68667225747764379254401762116, −1.97198077211416861539434520807,
1.05360544942746112783773231755, 2.21416939634295556955218188583, 2.99769289181205920909227961806, 3.83804117434463970003234380751, 4.57761075023519356061787537343, 6.16309728399489559298247429638, 6.62261830364806363565868760777, 7.53758556063866031248346791656, 7.950134946399620566929923618986, 8.651746573670143923419291260491