| L(s) = 1 | − 3-s − 4·7-s + 9-s + 11-s + 3·13-s + 17-s − 19-s + 4·21-s − 23-s − 27-s + 7·29-s + 9·31-s − 33-s + 4·37-s − 3·39-s − 6·41-s − 11·43-s + 9·49-s − 51-s + 6·53-s + 57-s − 4·59-s − 6·61-s − 4·63-s + 14·67-s + 69-s + 71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s + 0.301·11-s + 0.832·13-s + 0.242·17-s − 0.229·19-s + 0.872·21-s − 0.208·23-s − 0.192·27-s + 1.29·29-s + 1.61·31-s − 0.174·33-s + 0.657·37-s − 0.480·39-s − 0.937·41-s − 1.67·43-s + 9/7·49-s − 0.140·51-s + 0.824·53-s + 0.132·57-s − 0.520·59-s − 0.768·61-s − 0.503·63-s + 1.71·67-s + 0.120·69-s + 0.118·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.973586381\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.973586381\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 - 19 T + p T^{2} \) |
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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
−12.94587280263770, −12.45636909588239, −11.98303718570389, −11.72522867265497, −11.09283686534294, −10.53389909151602, −10.12797800498197, −9.789554710734174, −9.374967271953949, −8.606476311054103, −8.384135459519772, −7.750616920397640, −7.012617767995740, −6.564923552067340, −6.265217344346437, −6.041971816725649, −5.111536588995405, −4.819599953583409, −4.044165174985211, −3.568417065314183, −3.135763871705831, −2.505672467222931, −1.745961126591687, −0.8905491558930344, −0.5223635544899653,
0.5223635544899653, 0.8905491558930344, 1.745961126591687, 2.505672467222931, 3.135763871705831, 3.568417065314183, 4.044165174985211, 4.819599953583409, 5.111536588995405, 6.041971816725649, 6.265217344346437, 6.564923552067340, 7.012617767995740, 7.750616920397640, 8.384135459519772, 8.606476311054103, 9.374967271953949, 9.789554710734174, 10.12797800498197, 10.53389909151602, 11.09283686534294, 11.72522867265497, 11.98303718570389, 12.45636909588239, 12.94587280263770
