L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s + 5·11-s − 12-s + 14-s + 16-s − 17-s − 18-s + 21-s − 5·22-s − 4·23-s + 24-s − 27-s − 28-s − 3·29-s + 31-s − 32-s − 5·33-s + 34-s + 36-s + 37-s − 41-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 1.50·11-s − 0.288·12-s + 0.267·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.218·21-s − 1.06·22-s − 0.834·23-s + 0.204·24-s − 0.192·27-s − 0.188·28-s − 0.557·29-s + 0.179·31-s − 0.176·32-s − 0.870·33-s + 0.171·34-s + 1/6·36-s + 0.164·37-s − 0.156·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 37 | \( 1 - T \) |
good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−7.76340503484730031569689355600, −7.03541764490391818783975365426, −6.36672505348087401125042716826, −5.98795234899382904083345511724, −4.90876831083795781067824295603, −4.03120336841708847595305509720, −3.29888373701359386532052947393, −2.04764477778714175548258706268, −1.21069753372676115578743296579, 0,
1.21069753372676115578743296579, 2.04764477778714175548258706268, 3.29888373701359386532052947393, 4.03120336841708847595305509720, 4.90876831083795781067824295603, 5.98795234899382904083345511724, 6.36672505348087401125042716826, 7.03541764490391818783975365426, 7.76340503484730031569689355600