| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s − 2·9-s + 10-s − 12-s − 13-s − 14-s + 15-s + 16-s − 4·17-s + 2·18-s + 3·19-s − 20-s − 21-s − 3·23-s + 24-s + 25-s + 26-s + 5·27-s + 28-s − 2·29-s − 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s − 2/3·9-s + 0.316·10-s − 0.288·12-s − 0.277·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.970·17-s + 0.471·18-s + 0.688·19-s − 0.223·20-s − 0.218·21-s − 0.625·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.962·27-s + 0.188·28-s − 0.371·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 17 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.43977838426418509677024959461, −6.91362110222841085867832420609, −6.06632732212645623957778244129, −5.52277834404263372287665020364, −4.69366028765107046417484787869, −3.92300220466333541478680720281, −2.88512048103752100830034071054, −2.14911084296292626797635585340, −0.958879211022868015863565762622, 0,
0.958879211022868015863565762622, 2.14911084296292626797635585340, 2.88512048103752100830034071054, 3.92300220466333541478680720281, 4.69366028765107046417484787869, 5.52277834404263372287665020364, 6.06632732212645623957778244129, 6.91362110222841085867832420609, 7.43977838426418509677024959461
