L(s) = 1 | + 3·3-s + 6·5-s − 20·7-s + 9·9-s − 24·11-s + 13·13-s + 18·15-s − 30·17-s + 16·19-s − 60·21-s + 72·23-s − 89·25-s + 27·27-s − 282·29-s − 164·31-s − 72·33-s − 120·35-s + 110·37-s + 39·39-s − 126·41-s − 164·43-s + 54·45-s + 204·47-s + 57·49-s − 90·51-s − 738·53-s − 144·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.536·5-s − 1.07·7-s + 1/3·9-s − 0.657·11-s + 0.277·13-s + 0.309·15-s − 0.428·17-s + 0.193·19-s − 0.623·21-s + 0.652·23-s − 0.711·25-s + 0.192·27-s − 1.80·29-s − 0.950·31-s − 0.379·33-s − 0.579·35-s + 0.488·37-s + 0.160·39-s − 0.479·41-s − 0.581·43-s + 0.178·45-s + 0.633·47-s + 0.166·49-s − 0.247·51-s − 1.91·53-s − 0.353·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 13 | \( 1 - p T \) |
good | 5 | \( 1 - 6 T + p^{3} T^{2} \) |
| 7 | \( 1 + 20 T + p^{3} T^{2} \) |
| 11 | \( 1 + 24 T + p^{3} T^{2} \) |
| 17 | \( 1 + 30 T + p^{3} T^{2} \) |
| 19 | \( 1 - 16 T + p^{3} T^{2} \) |
| 23 | \( 1 - 72 T + p^{3} T^{2} \) |
| 29 | \( 1 + 282 T + p^{3} T^{2} \) |
| 31 | \( 1 + 164 T + p^{3} T^{2} \) |
| 37 | \( 1 - 110 T + p^{3} T^{2} \) |
| 41 | \( 1 + 126 T + p^{3} T^{2} \) |
| 43 | \( 1 + 164 T + p^{3} T^{2} \) |
| 47 | \( 1 - 204 T + p^{3} T^{2} \) |
| 53 | \( 1 + 738 T + p^{3} T^{2} \) |
| 59 | \( 1 + 120 T + p^{3} T^{2} \) |
| 61 | \( 1 - 614 T + p^{3} T^{2} \) |
| 67 | \( 1 + 848 T + p^{3} T^{2} \) |
| 71 | \( 1 + 132 T + p^{3} T^{2} \) |
| 73 | \( 1 - 218 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1096 T + p^{3} T^{2} \) |
| 83 | \( 1 + 552 T + p^{3} T^{2} \) |
| 89 | \( 1 - 210 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1726 T + p^{3} T^{2} \) |
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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
−9.546900210821679286700026770348, −9.187688498005841625032270074148, −8.011040252800378376925797903022, −7.12174286609688391661393449228, −6.17076849160542616540780103231, −5.25619123462550059734827206700, −3.85135646283813820969142191764, −2.94088293512803917780927237310, −1.80684859441965568236244382169, 0,
1.80684859441965568236244382169, 2.94088293512803917780927237310, 3.85135646283813820969142191764, 5.25619123462550059734827206700, 6.17076849160542616540780103231, 7.12174286609688391661393449228, 8.011040252800378376925797903022, 9.187688498005841625032270074148, 9.546900210821679286700026770348