L(s) = 1 | − 2-s − 3-s + 4-s − 3·5-s + 6-s − 8-s + 9-s + 3·10-s − 2·11-s − 12-s + 3·13-s + 3·15-s + 16-s − 3·17-s − 18-s − 3·20-s + 2·22-s − 23-s + 24-s + 4·25-s − 3·26-s − 27-s − 8·29-s − 3·30-s + 10·31-s − 32-s + 2·33-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.948·10-s − 0.603·11-s − 0.288·12-s + 0.832·13-s + 0.774·15-s + 1/4·16-s − 0.727·17-s − 0.235·18-s − 0.670·20-s + 0.426·22-s − 0.208·23-s + 0.204·24-s + 4/5·25-s − 0.588·26-s − 0.192·27-s − 1.48·29-s − 0.547·30-s + 1.79·31-s − 0.176·32-s + 0.348·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−7.74752568279194150371002963455, −7.07279367485576323223219873025, −6.34733778952269237197122686368, −5.65193456884718336394870220977, −4.66347487479745806024402998603, −4.01444114483548831071281583032, −3.21116612198115003948206718442, −2.14637571013113553823315376918, −0.917606406078886016312881753757, 0,
0.917606406078886016312881753757, 2.14637571013113553823315376918, 3.21116612198115003948206718442, 4.01444114483548831071281583032, 4.66347487479745806024402998603, 5.65193456884718336394870220977, 6.34733778952269237197122686368, 7.07279367485576323223219873025, 7.74752568279194150371002963455