L(s) = 1 | − 3-s + 2·5-s − 4·7-s + 9-s + 13-s − 2·15-s + 17-s + 4·19-s + 4·21-s − 25-s − 27-s + 6·29-s + 4·31-s − 8·35-s − 6·37-s − 39-s − 2·41-s − 4·43-s + 2·45-s + 9·49-s − 51-s − 10·53-s − 4·57-s − 4·59-s + 14·61-s − 4·63-s + 2·65-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.894·5-s − 1.51·7-s + 1/3·9-s + 0.277·13-s − 0.516·15-s + 0.242·17-s + 0.917·19-s + 0.872·21-s − 1/5·25-s − 0.192·27-s + 1.11·29-s + 0.718·31-s − 1.35·35-s − 0.986·37-s − 0.160·39-s − 0.312·41-s − 0.609·43-s + 0.298·45-s + 9/7·49-s − 0.140·51-s − 1.37·53-s − 0.529·57-s − 0.520·59-s + 1.79·61-s − 0.503·63-s + 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.457987312\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.457987312\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−8.207753182161725750902231108356, −7.24940346992202933671371603096, −6.52593098189616756590027884176, −6.15440652576650348219117830680, −5.44519585054997860351000139990, −4.67783381622893736169789755153, −3.52502676939184480898209721002, −2.96297847544688478831558152521, −1.81341721379023192715981808954, −0.66992401977953517979344965164,
0.66992401977953517979344965164, 1.81341721379023192715981808954, 2.96297847544688478831558152521, 3.52502676939184480898209721002, 4.67783381622893736169789755153, 5.44519585054997860351000139990, 6.15440652576650348219117830680, 6.52593098189616756590027884176, 7.24940346992202933671371603096, 8.207753182161725750902231108356