| L(s) = 1 | + 2·7-s − 4·11-s − 6·13-s − 6·17-s − 4·23-s − 5·25-s + 4·29-s + 10·31-s − 2·37-s + 2·41-s − 8·43-s + 12·47-s − 3·49-s − 12·53-s − 4·59-s − 2·61-s − 4·67-s + 4·71-s − 10·73-s − 8·77-s − 6·79-s + 12·83-s − 2·89-s − 12·91-s − 6·97-s + 4·101-s − 10·103-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 1.20·11-s − 1.66·13-s − 1.45·17-s − 0.834·23-s − 25-s + 0.742·29-s + 1.79·31-s − 0.328·37-s + 0.312·41-s − 1.21·43-s + 1.75·47-s − 3/7·49-s − 1.64·53-s − 0.520·59-s − 0.256·61-s − 0.488·67-s + 0.474·71-s − 1.17·73-s − 0.911·77-s − 0.675·79-s + 1.31·83-s − 0.211·89-s − 1.25·91-s − 0.609·97-s + 0.398·101-s − 0.985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−9.514926545655033706616286267341, −8.352864246059164846839031639199, −7.86636283231227873209414758068, −7.00186001179257888349709337610, −5.96427857852485389909948133454, −4.86024035915155668889978626292, −4.46736447254899856037450817253, −2.79778517840079752431830788770, −2.02071693225675944817521065685, 0,
2.02071693225675944817521065685, 2.79778517840079752431830788770, 4.46736447254899856037450817253, 4.86024035915155668889978626292, 5.96427857852485389909948133454, 7.00186001179257888349709337610, 7.86636283231227873209414758068, 8.352864246059164846839031639199, 9.514926545655033706616286267341
