| L(s) = 1 | − 3-s + 2·5-s + 9-s + 4·11-s + 2·13-s − 2·15-s + 17-s − 4·19-s − 25-s − 27-s + 10·29-s + 8·31-s − 4·33-s + 2·37-s − 2·39-s + 10·41-s − 12·43-s + 2·45-s − 7·49-s − 51-s − 6·53-s + 8·55-s + 4·57-s − 12·59-s + 10·61-s + 4·65-s + 12·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1/3·9-s + 1.20·11-s + 0.554·13-s − 0.516·15-s + 0.242·17-s − 0.917·19-s − 1/5·25-s − 0.192·27-s + 1.85·29-s + 1.43·31-s − 0.696·33-s + 0.328·37-s − 0.320·39-s + 1.56·41-s − 1.82·43-s + 0.298·45-s − 49-s − 0.140·51-s − 0.824·53-s + 1.07·55-s + 0.529·57-s − 1.56·59-s + 1.28·61-s + 0.496·65-s + 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.092580593\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.092580593\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−8.596549626610906858679281484900, −8.070372649571749969977602460565, −6.69848386214369198167777203066, −6.46056135767183933567189521369, −5.81286052920193080307583986322, −4.79105665208598685854610251022, −4.13815320087610274634408079960, −3.00991765579318910790552780461, −1.85270319908183405578953120836, −0.953921632956008845825292771843,
0.953921632956008845825292771843, 1.85270319908183405578953120836, 3.00991765579318910790552780461, 4.13815320087610274634408079960, 4.79105665208598685854610251022, 5.81286052920193080307583986322, 6.46056135767183933567189521369, 6.69848386214369198167777203066, 8.070372649571749969977602460565, 8.596549626610906858679281484900
