L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s − i·7-s + 0.999·8-s + (0.499 + 0.866i)10-s + (−0.866 − 1.5i)11-s + 1.73·13-s + (0.866 + 0.5i)14-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)19-s − 0.999·20-s + 1.73·22-s + (−0.5 + 0.866i)23-s + (−0.499 − 0.866i)25-s + (−0.866 + 1.49i)26-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s − i·7-s + 0.999·8-s + (0.499 + 0.866i)10-s + (−0.866 − 1.5i)11-s + 1.73·13-s + (0.866 + 0.5i)14-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)19-s − 0.999·20-s + 1.73·22-s + (−0.5 + 0.866i)23-s + (−0.499 − 0.866i)25-s + (−0.866 + 1.49i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8987987165\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8987987165\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + iT \) |
good | 11 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 - 1.73T + T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + 1.73T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 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
−8.708202029244999228751774035617, −8.275371553223509426037140250954, −7.69094342734991405660222969710, −6.57353155962901573915586939451, −5.79523984363789229939104948339, −5.50491354071707484736288524031, −4.24537433939005443392608169263, −3.56123711740943228490597015783, −1.70773828801080606897512908778, −0.73153284600720958533665269965,
1.71264965498424671970413934753, 2.44368414825167346409632343769, 3.16709552451382687345996459902, 4.31932665036546984463636217158, 5.18752993853505047890348709297, 6.28154495251464900403373981457, 6.90793447791917104662915218013, 7.975990845762514812652408472729, 8.549790136339302155564244480718, 9.332408748258490948486748361630