L(s) = 1 | − 3-s + 7-s − 2·9-s + 3·11-s + 2·13-s − 3·17-s + 5·19-s − 21-s − 23-s + 5·27-s − 6·29-s + 4·31-s − 3·33-s − 4·37-s − 2·39-s + 3·41-s − 8·43-s − 6·47-s + 49-s + 3·51-s + 12·53-s − 5·57-s − 12·59-s − 2·61-s − 2·63-s + 7·67-s + 69-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.377·7-s − 2/3·9-s + 0.904·11-s + 0.554·13-s − 0.727·17-s + 1.14·19-s − 0.218·21-s − 0.208·23-s + 0.962·27-s − 1.11·29-s + 0.718·31-s − 0.522·33-s − 0.657·37-s − 0.320·39-s + 0.468·41-s − 1.21·43-s − 0.875·47-s + 1/7·49-s + 0.420·51-s + 1.64·53-s − 0.662·57-s − 1.56·59-s − 0.256·61-s − 0.251·63-s + 0.855·67-s + 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 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
−13.17830777114655, −12.30522879603721, −12.09385737751610, −11.68117398546276, −11.14136126702805, −11.00278665890762, −10.41718562589687, −9.713857504155500, −9.381906088690319, −8.887252041335988, −8.354190342098398, −8.029008973912963, −7.351308164612734, −6.742621225621290, −6.504149280411822, −5.842177803854984, −5.487454414058888, −4.918678007659673, −4.491170718083199, −3.656186851997780, −3.487651130949662, −2.664682421611004, −2.014301222049823, −1.379096854389854, −0.7919174265737701, 0,
0.7919174265737701, 1.379096854389854, 2.014301222049823, 2.664682421611004, 3.487651130949662, 3.656186851997780, 4.491170718083199, 4.918678007659673, 5.487454414058888, 5.842177803854984, 6.504149280411822, 6.742621225621290, 7.351308164612734, 8.029008973912963, 8.354190342098398, 8.887252041335988, 9.381906088690319, 9.713857504155500, 10.41718562589687, 11.00278665890762, 11.14136126702805, 11.68117398546276, 12.09385737751610, 12.30522879603721, 13.17830777114655