L(s) = 1 | + 3-s − 4·7-s + 9-s − 4·11-s − 13-s + 6·17-s − 4·21-s − 8·23-s + 27-s + 6·29-s + 4·31-s − 4·33-s + 2·37-s − 39-s − 10·41-s − 4·43-s − 8·47-s + 9·49-s + 6·51-s + 2·53-s − 12·59-s − 2·61-s − 4·63-s − 16·67-s − 8·69-s + 8·71-s + 6·73-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.51·7-s + 1/3·9-s − 1.20·11-s − 0.277·13-s + 1.45·17-s − 0.872·21-s − 1.66·23-s + 0.192·27-s + 1.11·29-s + 0.718·31-s − 0.696·33-s + 0.328·37-s − 0.160·39-s − 1.56·41-s − 0.609·43-s − 1.16·47-s + 9/7·49-s + 0.840·51-s + 0.274·53-s − 1.56·59-s − 0.256·61-s − 0.503·63-s − 1.95·67-s − 0.963·69-s + 0.949·71-s + 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.314801790\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.314801790\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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
−15.93766497672428, −15.50274205988605, −15.02693197849005, −14.18732650977224, −13.75255327317897, −13.29865044025376, −12.67944173009750, −12.14935267507006, −11.81156703275849, −10.56427038462744, −10.17924169714746, −9.881955783332556, −9.300125051872818, −8.411561051519172, −7.921966083419844, −7.494432998145552, −6.482573947532576, −6.242270921542251, −5.323895540494503, −4.700233558183015, −3.697111778851675, −3.168151461239880, −2.686568627608191, −1.732493315693098, −0.4603849782038834,
0.4603849782038834, 1.732493315693098, 2.686568627608191, 3.168151461239880, 3.697111778851675, 4.700233558183015, 5.323895540494503, 6.242270921542251, 6.482573947532576, 7.494432998145552, 7.921966083419844, 8.411561051519172, 9.300125051872818, 9.881955783332556, 10.17924169714746, 10.56427038462744, 11.81156703275849, 12.14935267507006, 12.67944173009750, 13.29865044025376, 13.75255327317897, 14.18732650977224, 15.02693197849005, 15.50274205988605, 15.93766497672428