L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 2·7-s − 8-s + 9-s + 2·11-s − 12-s + 13-s + 2·14-s + 16-s − 2·17-s − 18-s − 4·19-s + 2·21-s − 2·22-s + 24-s − 26-s − 27-s − 2·28-s + 4·29-s + 8·31-s − 32-s − 2·33-s + 2·34-s + 36-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 + 0.603·11-s − 0.288·12-s + 0.277·13-s + 0.534·14-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.917·19-s + 0.436·21-s − 0.426·22-s + 0.204·24-s − 0.196·26-s − 0.192·27-s − 0.377·28-s + 0.742·29-s + 1.43·31-s − 0.176·32-s − 0.348·33-s + 0.342·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−8.762871783397594846228710724154, −8.232545338410278018445733038195, −6.96806663573867713351809691671, −6.59817683928018347550910114836, −5.88315580002259313007948255002, −4.72648356099096059668505397829, −3.76272059018480964618638483975, −2.63814879043714031851210280367, −1.35840044681910597035006671653, 0,
1.35840044681910597035006671653, 2.63814879043714031851210280367, 3.76272059018480964618638483975, 4.72648356099096059668505397829, 5.88315580002259313007948255002, 6.59817683928018347550910114836, 6.96806663573867713351809691671, 8.232545338410278018445733038195, 8.762871783397594846228710724154