| L(s) = 1 | − 3-s − 4·7-s + 9-s − 11-s − 4·13-s + 17-s − 4·19-s + 4·21-s − 8·23-s − 27-s − 4·29-s − 6·31-s + 33-s − 4·37-s + 4·39-s − 10·41-s + 4·43-s − 2·47-s + 9·49-s − 51-s + 4·53-s + 4·57-s − 4·59-s − 14·61-s − 4·63-s − 4·67-s + 8·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s − 0.301·11-s − 1.10·13-s + 0.242·17-s − 0.917·19-s + 0.872·21-s − 1.66·23-s − 0.192·27-s − 0.742·29-s − 1.07·31-s + 0.174·33-s − 0.657·37-s + 0.640·39-s − 1.56·41-s + 0.609·43-s − 0.291·47-s + 9/7·49-s − 0.140·51-s + 0.549·53-s + 0.529·57-s − 0.520·59-s − 1.79·61-s − 0.503·63-s − 0.488·67-s + 0.963·69-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−13.28376226417982, −12.61226494984849, −12.19440752654640, −12.05043910427983, −11.38411043520219, −10.70138322862786, −10.32088407277711, −10.01089509807721, −9.588589551117610, −9.032769534698795, −8.588966930414425, −7.859490917947918, −7.316072262114464, −7.065688414609253, −6.435584980384824, −5.843854499930440, −5.753221403520452, −4.988296170636766, −4.319831263194002, −3.989783628261768, −3.173033190115332, −2.912649921821821, −1.958191959295181, −1.680501070622942, −0.3419774503408602, 0,
0.3419774503408602, 1.680501070622942, 1.958191959295181, 2.912649921821821, 3.173033190115332, 3.989783628261768, 4.319831263194002, 4.988296170636766, 5.753221403520452, 5.843854499930440, 6.435584980384824, 7.065688414609253, 7.316072262114464, 7.859490917947918, 8.588966930414425, 9.032769534698795, 9.588589551117610, 10.01089509807721, 10.32088407277711, 10.70138322862786, 11.38411043520219, 12.05043910427983, 12.19440752654640, 12.61226494984849, 13.28376226417982
