L(s) = 1 | + 7-s − 13-s + 19-s + 25-s + 31-s − 37-s + 43-s + 2·61-s − 2·67-s + 2·73-s + 79-s − 91-s − 97-s − 2·103-s − 109-s + ⋯ |
L(s) = 1 | + 7-s − 13-s + 19-s + 25-s + 31-s − 37-s + 43-s + 2·61-s − 2·67-s + 2·73-s + 79-s − 91-s − 97-s − 2·103-s − 109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.462616976\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.462616976\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( 1 - T + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( 1 - T + T^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 - T + T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( 1 - T + T^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )^{2} \) |
| 67 | \( ( 1 + T )^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )^{2} \) |
| 79 | \( 1 - T + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( 1 + T + T^{2} \) |
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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
−8.558292672596323373346961270413, −7.922257766135825167903086962117, −7.25720700514919357024417312068, −6.57725486868969188476796962579, −5.42580987433040655593274686232, −4.99755673438722514495874419810, −4.20288316798077964235823012755, −3.09476470101742590814381015237, −2.22781392158689212425309689634, −1.09896846722295642777352785451,
1.09896846722295642777352785451, 2.22781392158689212425309689634, 3.09476470101742590814381015237, 4.20288316798077964235823012755, 4.99755673438722514495874419810, 5.42580987433040655593274686232, 6.57725486868969188476796962579, 7.25720700514919357024417312068, 7.922257766135825167903086962117, 8.558292672596323373346961270413