L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 2·7-s + 8-s + 9-s − 12-s − 2·13-s − 2·14-s + 16-s + 17-s + 18-s − 4·19-s + 2·21-s + 6·23-s − 24-s − 2·26-s − 27-s − 2·28-s − 10·31-s + 32-s + 34-s + 36-s − 8·37-s − 4·38-s + 2·39-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.288·12-s − 0.554·13-s − 0.534·14-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 0.917·19-s + 0.436·21-s + 1.25·23-s − 0.204·24-s − 0.392·26-s − 0.192·27-s − 0.377·28-s − 1.79·31-s + 0.176·32-s + 0.171·34-s + 1/6·36-s − 1.31·37-s − 0.648·38-s + 0.320·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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
−8.516143136114101940727160697764, −7.39776427066059355932306977819, −6.87962715487262458484700072265, −6.12836006413759304604077604546, −5.36248307034645218045013289720, −4.64968151044060984146183527578, −3.68664994402283801524702535725, −2.86146379401405857148189722343, −1.65383172558907138899083834410, 0,
1.65383172558907138899083834410, 2.86146379401405857148189722343, 3.68664994402283801524702535725, 4.64968151044060984146183527578, 5.36248307034645218045013289720, 6.12836006413759304604077604546, 6.87962715487262458484700072265, 7.39776427066059355932306977819, 8.516143136114101940727160697764