| L(s) = 1 | + 2·3-s − 5-s + 9-s − 2·11-s − 2·15-s + 2·17-s − 2·19-s + 2·23-s + 25-s − 4·27-s − 6·29-s − 2·31-s − 4·33-s + 6·37-s − 2·41-s + 6·43-s − 45-s + 8·47-s − 7·49-s + 4·51-s − 2·53-s + 2·55-s − 4·57-s − 6·59-s − 14·61-s + 4·69-s − 10·71-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s + 1/3·9-s − 0.603·11-s − 0.516·15-s + 0.485·17-s − 0.458·19-s + 0.417·23-s + 1/5·25-s − 0.769·27-s − 1.11·29-s − 0.359·31-s − 0.696·33-s + 0.986·37-s − 0.312·41-s + 0.914·43-s − 0.149·45-s + 1.16·47-s − 49-s + 0.560·51-s − 0.274·53-s + 0.269·55-s − 0.529·57-s − 0.781·59-s − 1.79·61-s + 0.481·69-s − 1.18·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.65145326531316609749473144008, −7.33387504597643637570519717846, −6.20297819498089116422846262337, −5.52292241535810330877601261053, −4.57569787487438471292926229119, −3.85287056779756280024412088003, −3.08843563443615203217480631996, −2.50392315128847178535779020476, −1.48618888866608499865943758358, 0,
1.48618888866608499865943758358, 2.50392315128847178535779020476, 3.08843563443615203217480631996, 3.85287056779756280024412088003, 4.57569787487438471292926229119, 5.52292241535810330877601261053, 6.20297819498089116422846262337, 7.33387504597643637570519717846, 7.65145326531316609749473144008
