L(s) = 1 | + 2-s + 4-s + 7-s + 8-s − 4·11-s − 2·13-s + 14-s + 16-s + 6·17-s − 4·22-s + 23-s − 2·26-s + 28-s + 2·29-s + 4·31-s + 32-s + 6·34-s − 6·37-s + 6·41-s − 12·43-s − 4·44-s + 46-s − 12·47-s + 49-s − 2·52-s + 6·53-s + 56-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s − 1.20·11-s − 0.554·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.852·22-s + 0.208·23-s − 0.392·26-s + 0.188·28-s + 0.371·29-s + 0.718·31-s + 0.176·32-s + 1.02·34-s − 0.986·37-s + 0.937·41-s − 1.82·43-s − 0.603·44-s + 0.147·46-s − 1.75·47-s + 1/7·49-s − 0.277·52-s + 0.824·53-s + 0.133·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−14.38543520541491, −13.82417880029099, −13.39972587985485, −12.84262693527397, −12.41454282527449, −11.90007700892863, −11.53597958688538, −10.82134829659624, −10.37070204619277, −9.950142488586063, −9.477512060423580, −8.521618567567872, −8.106517622565210, −7.734632527480819, −7.079149947765393, −6.614710433538812, −5.823086324590405, −5.358612083285335, −4.934422381335114, −4.476815516391392, −3.551074196384802, −3.119976262179713, −2.527235969028953, −1.804955417925684, −1.046876136723577, 0,
1.046876136723577, 1.804955417925684, 2.527235969028953, 3.119976262179713, 3.551074196384802, 4.476815516391392, 4.934422381335114, 5.358612083285335, 5.823086324590405, 6.614710433538812, 7.079149947765393, 7.734632527480819, 8.106517622565210, 8.521618567567872, 9.477512060423580, 9.950142488586063, 10.37070204619277, 10.82134829659624, 11.53597958688538, 11.90007700892863, 12.41454282527449, 12.84262693527397, 13.39972587985485, 13.82417880029099, 14.38543520541491