L(s) = 1 | + 2·2-s − 3-s + 2·4-s − 5-s − 2·6-s + 2·7-s + 9-s − 2·10-s + 3·11-s − 2·12-s + 3·13-s + 4·14-s + 15-s − 4·16-s + 17-s + 2·18-s + 19-s − 2·20-s − 2·21-s + 6·22-s − 7·23-s − 4·25-s + 6·26-s − 27-s + 4·28-s − 2·29-s + 2·30-s + ⋯ |
L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s − 0.447·5-s − 0.816·6-s + 0.755·7-s + 1/3·9-s − 0.632·10-s + 0.904·11-s − 0.577·12-s + 0.832·13-s + 1.06·14-s + 0.258·15-s − 16-s + 0.242·17-s + 0.471·18-s + 0.229·19-s − 0.447·20-s − 0.436·21-s + 1.27·22-s − 1.45·23-s − 4/5·25-s + 1.17·26-s − 0.192·27-s + 0.755·28-s − 0.371·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| 53 | \( 1 + T \) |
good | 2 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 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
−14.00142162515245, −13.66962678089508, −13.00465651726595, −12.49624506959137, −11.90725325814974, −11.79988443851420, −11.36649457301913, −10.81195006445202, −10.30138748714382, −9.524832499037356, −9.099955791928274, −8.422627715996364, −7.785189938367988, −7.501535642468895, −6.499642309846511, −6.323211604990791, −5.839821333318269, −5.167089079160946, −4.785156868729121, −4.041826577958456, −3.848005368634488, −3.335403628845769, −2.335363561446685, −1.756924577160709, −1.008374391004073, 0,
1.008374391004073, 1.756924577160709, 2.335363561446685, 3.335403628845769, 3.848005368634488, 4.041826577958456, 4.785156868729121, 5.167089079160946, 5.839821333318269, 6.323211604990791, 6.499642309846511, 7.501535642468895, 7.785189938367988, 8.422627715996364, 9.099955791928274, 9.524832499037356, 10.30138748714382, 10.81195006445202, 11.36649457301913, 11.79988443851420, 11.90725325814974, 12.49624506959137, 13.00465651726595, 13.66962678089508, 14.00142162515245