L(s) = 1 | − 3-s − 2·4-s − 7-s + 9-s − 11-s + 2·12-s − 3·13-s + 4·16-s − 6·17-s + 21-s + 23-s − 5·25-s − 27-s + 2·28-s − 4·29-s − 4·31-s + 33-s − 2·36-s − 7·37-s + 3·39-s − 3·41-s + 6·43-s + 2·44-s − 4·47-s − 4·48-s + 49-s + 6·51-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 4-s − 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.577·12-s − 0.832·13-s + 16-s − 1.45·17-s + 0.218·21-s + 0.208·23-s − 25-s − 0.192·27-s + 0.377·28-s − 0.742·29-s − 0.718·31-s + 0.174·33-s − 1/3·36-s − 1.15·37-s + 0.480·39-s − 0.468·41-s + 0.914·43-s + 0.301·44-s − 0.583·47-s − 0.577·48-s + 1/7·49-s + 0.840·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 83391 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83391 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−14.09916751769685, −13.64453078204545, −13.13596464738469, −12.71732420950415, −12.42477289178962, −11.70121428836257, −11.21797321327570, −10.71790759373552, −10.12125910367610, −9.710885972710065, −9.219216433970731, −8.785347617980574, −8.173518189602550, −7.495851727381843, −7.143921008802477, −6.407236108084551, −5.920069604745131, −5.240077752744684, −4.924560280456584, −4.289384352037776, −3.766685520391646, −3.141854058468645, −2.222383415567251, −1.679597031288478, −0.5204284459246882, 0,
0.5204284459246882, 1.679597031288478, 2.222383415567251, 3.141854058468645, 3.766685520391646, 4.289384352037776, 4.924560280456584, 5.240077752744684, 5.920069604745131, 6.407236108084551, 7.143921008802477, 7.495851727381843, 8.173518189602550, 8.785347617980574, 9.219216433970731, 9.710885972710065, 10.12125910367610, 10.71790759373552, 11.21797321327570, 11.70121428836257, 12.42477289178962, 12.71732420950415, 13.13596464738469, 13.64453078204545, 14.09916751769685