L(s) = 1 | + 2-s + 4-s − 5-s + 8-s − 10-s − 13-s + 16-s − 17-s − 20-s − 26-s − 29-s + 32-s − 34-s − 37-s − 40-s + 2·41-s + 49-s − 52-s + 2·53-s − 58-s − 61-s + 64-s + 65-s − 68-s − 73-s − 74-s − 80-s + ⋯ |
L(s) = 1 | + 2-s + 4-s − 5-s + 8-s − 10-s − 13-s + 16-s − 17-s − 20-s − 26-s − 29-s + 32-s − 34-s − 37-s − 40-s + 2·41-s + 49-s − 52-s + 2·53-s − 58-s − 61-s + 64-s + 65-s − 68-s − 73-s − 74-s − 80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.188200343\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.188200343\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + T + T^{2} \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( 1 + T + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( ( 1 - T )^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( 1 + T + T^{2} \) |
| 97 | \( ( 1 - T )^{2} \) |
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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
−11.92087871252312978719466341079, −11.22600475906675974031770849257, −10.31038993583302778273367538430, −8.954948356116674124248041963203, −7.65216242953380728411403903154, −7.10118567986892139403294393430, −5.80122382318090507880480519818, −4.62590085768480636130122278343, −3.78882632007711462473940508430, −2.39124768407718997212617200088,
2.39124768407718997212617200088, 3.78882632007711462473940508430, 4.62590085768480636130122278343, 5.80122382318090507880480519818, 7.10118567986892139403294393430, 7.65216242953380728411403903154, 8.954948356116674124248041963203, 10.31038993583302778273367538430, 11.22600475906675974031770849257, 11.92087871252312978719466341079