| L(s) = 1 | − 5-s − 2·7-s + 13-s − 23-s + 25-s + 3·29-s + 3·31-s + 2·35-s − 8·37-s − 3·41-s − 2·43-s + 11·47-s − 3·49-s + 14·53-s + 8·59-s − 4·61-s − 65-s − 4·67-s − 7·71-s − 9·73-s − 4·83-s + 2·89-s − 2·91-s + 18·97-s − 18·101-s − 4·103-s + 16·107-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s + 0.277·13-s − 0.208·23-s + 1/5·25-s + 0.557·29-s + 0.538·31-s + 0.338·35-s − 1.31·37-s − 0.468·41-s − 0.304·43-s + 1.60·47-s − 3/7·49-s + 1.92·53-s + 1.04·59-s − 0.512·61-s − 0.124·65-s − 0.488·67-s − 0.830·71-s − 1.05·73-s − 0.439·83-s + 0.211·89-s − 0.209·91-s + 1.82·97-s − 1.79·101-s − 0.394·103-s + 1.54·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−7.30642357360754097422780035267, −6.88567717024607840778405721075, −6.10438894301055586828236499588, −5.45058531291258515824418141205, −4.55727891129819166463808143570, −3.83285118546953039314820695012, −3.16742016578203370753833638372, −2.33359787398909904098188020043, −1.13836798656758523790560761948, 0,
1.13836798656758523790560761948, 2.33359787398909904098188020043, 3.16742016578203370753833638372, 3.83285118546953039314820695012, 4.55727891129819166463808143570, 5.45058531291258515824418141205, 6.10438894301055586828236499588, 6.88567717024607840778405721075, 7.30642357360754097422780035267
