L(s) = 1 | − 3-s + 2·7-s + 9-s − 2·21-s − 25-s − 27-s + 37-s + 3·49-s + 2·63-s + 2·67-s − 2·73-s + 75-s + 81-s − 111-s + ⋯ |
L(s) = 1 | − 3-s + 2·7-s + 9-s − 2·21-s − 25-s − 27-s + 37-s + 3·49-s + 2·63-s + 2·67-s − 2·73-s + 75-s + 81-s − 111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.050052551\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.050052551\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 37 | \( 1 - T \) |
good | 5 | \( 1 + T^{2} \) |
| 7 | \( ( 1 - T )^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 + T )^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−9.607802282887666890290522230699, −8.544691620823811235250958133716, −7.82182909815563501771744016286, −7.21518932728387963466040801750, −6.10107227607268113720615972869, −5.37241650071437557940613373125, −4.67740826158460570736013309643, −3.97097573600309252851361150463, −2.19834462836226889317934222558, −1.22840974389690426951788623925,
1.22840974389690426951788623925, 2.19834462836226889317934222558, 3.97097573600309252851361150463, 4.67740826158460570736013309643, 5.37241650071437557940613373125, 6.10107227607268113720615972869, 7.21518932728387963466040801750, 7.82182909815563501771744016286, 8.544691620823811235250958133716, 9.607802282887666890290522230699