L(s) = 1 | − 3-s − 5-s + 2·7-s + 9-s + 2·11-s + 4·13-s + 15-s − 6·17-s − 8·19-s − 2·21-s + 4·23-s + 25-s − 27-s + 29-s + 10·31-s − 2·33-s − 2·35-s + 2·37-s − 4·39-s − 4·43-s − 45-s + 8·47-s − 3·49-s + 6·51-s + 6·53-s − 2·55-s + 8·57-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.603·11-s + 1.10·13-s + 0.258·15-s − 1.45·17-s − 1.83·19-s − 0.436·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s + 0.185·29-s + 1.79·31-s − 0.348·33-s − 0.338·35-s + 0.328·37-s − 0.640·39-s − 0.609·43-s − 0.149·45-s + 1.16·47-s − 3/7·49-s + 0.840·51-s + 0.824·53-s − 0.269·55-s + 1.05·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.616853879\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.616853879\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 - T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.112824149912027512674309313415, −7.07320013292000895891999543151, −6.49478861024419837314147789967, −6.05849102829246774337392932138, −4.93043689814539626411171853871, −4.38407431404875458834223796691, −3.88636691880150337358171216919, −2.64014909665642074431197474843, −1.67975062704864302370618628001, −0.69189405861742330592153584116,
0.69189405861742330592153584116, 1.67975062704864302370618628001, 2.64014909665642074431197474843, 3.88636691880150337358171216919, 4.38407431404875458834223796691, 4.93043689814539626411171853871, 6.05849102829246774337392932138, 6.49478861024419837314147789967, 7.07320013292000895891999543151, 8.112824149912027512674309313415