| L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·6-s + 4·7-s − 8-s + 9-s − 2·12-s − 2·13-s − 4·14-s + 16-s + 17-s − 18-s + 4·19-s − 8·21-s + 2·24-s − 5·25-s + 2·26-s + 4·27-s + 4·28-s − 4·31-s − 32-s − 34-s + 36-s − 4·37-s − 4·38-s + 4·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.577·12-s − 0.554·13-s − 1.06·14-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 0.917·19-s − 1.74·21-s + 0.408·24-s − 25-s + 0.392·26-s + 0.769·27-s + 0.755·28-s − 0.718·31-s − 0.176·32-s − 0.171·34-s + 1/6·36-s − 0.657·37-s − 0.648·38-s + 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−8.076455834435977550787516164955, −7.36497483724733271440784650514, −6.72084452362773307312329845344, −5.67320794510931157328652310045, −5.26894992853838733045784795081, −4.56197836266380508635038039003, −3.34611668167428925701071579672, −2.05022911292484155356003327044, −1.25284776655083263545298048613, 0,
1.25284776655083263545298048613, 2.05022911292484155356003327044, 3.34611668167428925701071579672, 4.56197836266380508635038039003, 5.26894992853838733045784795081, 5.67320794510931157328652310045, 6.72084452362773307312329845344, 7.36497483724733271440784650514, 8.076455834435977550787516164955
