| L(s) = 1 | + 3·3-s − 5·5-s + 32·7-s + 9·9-s − 36·11-s + 10·13-s − 15·15-s − 78·17-s − 140·19-s + 96·21-s − 192·23-s + 25·25-s + 27·27-s − 6·29-s − 16·31-s − 108·33-s − 160·35-s + 34·37-s + 30·39-s − 390·41-s + 52·43-s − 45·45-s + 408·47-s + 681·49-s − 234·51-s + 114·53-s + 180·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.72·7-s + 1/3·9-s − 0.986·11-s + 0.213·13-s − 0.258·15-s − 1.11·17-s − 1.69·19-s + 0.997·21-s − 1.74·23-s + 1/5·25-s + 0.192·27-s − 0.0384·29-s − 0.0926·31-s − 0.569·33-s − 0.772·35-s + 0.151·37-s + 0.123·39-s − 1.48·41-s + 0.184·43-s − 0.149·45-s + 1.26·47-s + 1.98·49-s − 0.642·51-s + 0.295·53-s + 0.441·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 + p T \) |
| good | 7 | \( 1 - 32 T + p^{3} T^{2} \) |
| 11 | \( 1 + 36 T + p^{3} T^{2} \) |
| 13 | \( 1 - 10 T + p^{3} T^{2} \) |
| 17 | \( 1 + 78 T + p^{3} T^{2} \) |
| 19 | \( 1 + 140 T + p^{3} T^{2} \) |
| 23 | \( 1 + 192 T + p^{3} T^{2} \) |
| 29 | \( 1 + 6 T + p^{3} T^{2} \) |
| 31 | \( 1 + 16 T + p^{3} T^{2} \) |
| 37 | \( 1 - 34 T + p^{3} T^{2} \) |
| 41 | \( 1 + 390 T + p^{3} T^{2} \) |
| 43 | \( 1 - 52 T + p^{3} T^{2} \) |
| 47 | \( 1 - 408 T + p^{3} T^{2} \) |
| 53 | \( 1 - 114 T + p^{3} T^{2} \) |
| 59 | \( 1 + 516 T + p^{3} T^{2} \) |
| 61 | \( 1 - 58 T + p^{3} T^{2} \) |
| 67 | \( 1 - 892 T + p^{3} T^{2} \) |
| 71 | \( 1 + 120 T + p^{3} T^{2} \) |
| 73 | \( 1 + 646 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1168 T + p^{3} T^{2} \) |
| 83 | \( 1 - 732 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1590 T + p^{3} T^{2} \) |
| 97 | \( 1 - 2 p T + p^{3} T^{2} \) |
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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
−8.887826968178397856267549434622, −8.270211259524226652549588677898, −7.888867422601992085619798310678, −6.85152418192609713843253703913, −5.63724848521262893273036899118, −4.55498881771019083374099929415, −4.05491898491941535585693567812, −2.45783193774510891064477770644, −1.74924392437590458176167981534, 0,
1.74924392437590458176167981534, 2.45783193774510891064477770644, 4.05491898491941535585693567812, 4.55498881771019083374099929415, 5.63724848521262893273036899118, 6.85152418192609713843253703913, 7.888867422601992085619798310678, 8.270211259524226652549588677898, 8.887826968178397856267549434622
