| L(s) = 1 | + 3·7-s + 3·11-s + 13-s + 17-s − 19-s + 4·23-s − 5·29-s − 5·37-s − 5·41-s + 8·43-s − 5·47-s + 2·49-s + 3·53-s + 6·59-s + 4·61-s − 12·67-s + 17·73-s + 9·77-s + 4·79-s + 16·83-s − 18·89-s + 3·91-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 1.13·7-s + 0.904·11-s + 0.277·13-s + 0.242·17-s − 0.229·19-s + 0.834·23-s − 0.928·29-s − 0.821·37-s − 0.780·41-s + 1.21·43-s − 0.729·47-s + 2/7·49-s + 0.412·53-s + 0.781·59-s + 0.512·61-s − 1.46·67-s + 1.98·73-s + 1.02·77-s + 0.450·79-s + 1.75·83-s − 1.90·89-s + 0.314·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 397800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 397800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 5 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 17 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−12.61184499178734, −12.14096472376068, −11.70859107585659, −11.29913822078173, −10.97487552472978, −10.50229074525482, −10.01098734768094, −9.355689827137420, −9.041831974327795, −8.650728737992261, −8.030305488510295, −7.793756787012335, −7.118527131861336, −6.745048370532300, −6.268857189609891, −5.611494850191815, −5.167918456296390, −4.835723640272030, −4.091687759727778, −3.797903743306991, −3.234375021703308, −2.472944597441380, −1.935675575668698, −1.365875554152425, −0.9545904194405554, 0,
0.9545904194405554, 1.365875554152425, 1.935675575668698, 2.472944597441380, 3.234375021703308, 3.797903743306991, 4.091687759727778, 4.835723640272030, 5.167918456296390, 5.611494850191815, 6.268857189609891, 6.745048370532300, 7.118527131861336, 7.793756787012335, 8.030305488510295, 8.650728737992261, 9.041831974327795, 9.355689827137420, 10.01098734768094, 10.50229074525482, 10.97487552472978, 11.29913822078173, 11.70859107585659, 12.14096472376068, 12.61184499178734
