| L(s) = 1 | + 4·2-s + 9·3-s + 16·4-s − 94·5-s + 36·6-s + 188·7-s + 64·8-s + 81·9-s − 376·10-s − 740·11-s + 144·12-s − 462·13-s + 752·14-s − 846·15-s + 256·16-s + 138·17-s + 324·18-s − 722·19-s − 1.50e3·20-s + 1.69e3·21-s − 2.96e3·22-s − 1.46e3·23-s + 576·24-s + 5.71e3·25-s − 1.84e3·26-s + 729·27-s + 3.00e3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.68·5-s + 0.408·6-s + 1.45·7-s + 0.353·8-s + 1/3·9-s − 1.18·10-s − 1.84·11-s + 0.288·12-s − 0.758·13-s + 1.02·14-s − 0.970·15-s + 1/4·16-s + 0.115·17-s + 0.235·18-s − 0.458·19-s − 0.840·20-s + 0.837·21-s − 1.30·22-s − 0.577·23-s + 0.204·24-s + 1.82·25-s − 0.536·26-s + 0.192·27-s + 0.725·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 222 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 222 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 3 | \( 1 - p^{2} T \) |
| 37 | \( 1 + p^{2} T \) |
| good | 5 | \( 1 + 94 T + p^{5} T^{2} \) |
| 7 | \( 1 - 188 T + p^{5} T^{2} \) |
| 11 | \( 1 + 740 T + p^{5} T^{2} \) |
| 13 | \( 1 + 462 T + p^{5} T^{2} \) |
| 17 | \( 1 - 138 T + p^{5} T^{2} \) |
| 19 | \( 1 + 2 p^{2} T + p^{5} T^{2} \) |
| 23 | \( 1 + 1466 T + p^{5} T^{2} \) |
| 29 | \( 1 + 6190 T + p^{5} T^{2} \) |
| 31 | \( 1 - 2618 T + p^{5} T^{2} \) |
| 41 | \( 1 + 15954 T + p^{5} T^{2} \) |
| 43 | \( 1 + 20906 T + p^{5} T^{2} \) |
| 47 | \( 1 - 4420 T + p^{5} T^{2} \) |
| 53 | \( 1 - 1658 T + p^{5} T^{2} \) |
| 59 | \( 1 + 32274 T + p^{5} T^{2} \) |
| 61 | \( 1 - 2098 T + p^{5} T^{2} \) |
| 67 | \( 1 - 46556 T + p^{5} T^{2} \) |
| 71 | \( 1 - 468 p T + p^{5} T^{2} \) |
| 73 | \( 1 + 55442 T + p^{5} T^{2} \) |
| 79 | \( 1 + 25546 T + p^{5} T^{2} \) |
| 83 | \( 1 - 76920 T + p^{5} T^{2} \) |
| 89 | \( 1 + 15134 T + p^{5} T^{2} \) |
| 97 | \( 1 + 80654 T + p^{5} T^{2} \) |
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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
−11.15569162637414105564484151997, −10.26259616453286401268365049868, −8.354466897118058995567411678447, −7.921631230117232545504349038604, −7.22271618450325933952234588148, −5.18799099559791795307467909073, −4.52595451386638227453167870998, −3.34283532151911540064858389208, −2.05865933639659465880662637027, 0,
2.05865933639659465880662637027, 3.34283532151911540064858389208, 4.52595451386638227453167870998, 5.18799099559791795307467909073, 7.22271618450325933952234588148, 7.921631230117232545504349038604, 8.354466897118058995567411678447, 10.26259616453286401268365049868, 11.15569162637414105564484151997
