| L(s) = 1 | + 3-s − 5-s − 4·7-s + 9-s − 11-s − 4·13-s − 15-s − 6·17-s + 2·19-s − 4·21-s + 25-s + 27-s − 4·31-s − 33-s + 4·35-s − 10·37-s − 4·39-s − 4·43-s − 45-s + 12·47-s + 9·49-s − 6·51-s + 6·53-s + 55-s + 2·57-s + 12·59-s − 10·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s − 0.301·11-s − 1.10·13-s − 0.258·15-s − 1.45·17-s + 0.458·19-s − 0.872·21-s + 1/5·25-s + 0.192·27-s − 0.718·31-s − 0.174·33-s + 0.676·35-s − 1.64·37-s − 0.640·39-s − 0.609·43-s − 0.149·45-s + 1.75·47-s + 9/7·49-s − 0.840·51-s + 0.824·53-s + 0.134·55-s + 0.264·57-s + 1.56·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 660 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−9.978183703853620625557175039707, −9.238061894079419833722803309921, −8.510418078048314070850105018894, −7.22741549048532299546614518657, −6.87977465279522291645710913052, −5.54701550218547875334577656336, −4.30987323692753027270428495802, −3.30360101068721859971758002522, −2.35548221701648672137868113902, 0,
2.35548221701648672137868113902, 3.30360101068721859971758002522, 4.30987323692753027270428495802, 5.54701550218547875334577656336, 6.87977465279522291645710913052, 7.22741549048532299546614518657, 8.510418078048314070850105018894, 9.238061894079419833722803309921, 9.978183703853620625557175039707
